I think OpenAI deserve credit for adjusting their approach after the last math drop. I think Terrance Tao shares some credit here for galvanizing the response.
Including the relevant mathematicians and scientists before dumping this onto the public is sure to go a long way to relaxing fears in those communities. I'm not sure how they will deal with accusations of favoritism but at least they are demonstrating a willingness to participate in a process.
I hope the community can express some positivity that will encourage OpenAI and other foundation labs to continue with this collaborative approach.
Well said. Attribution is important, for obvious human reasons and to direct people to valuable context. But our names will fade fast from history, what matters is the journey of human knowledge.
> It is amazing what you can accomplish if you do not care who gets the credit.
> It is amazing what you can accomplish if you do not care who gets the credit.
-- Me, 2026
Jokes aside, this is a thought provoking quote. I do think it's easy to say as a well known figurehead, when the quote and other actions below you may easily be attributed to you, at least partially. But I do agree somewhat, and I think the context around Truman's time (WWII) is a good example of what humanity can accomplish when the stakes at hand supersede the considerations of workers' rights or credit/ownership. Of course this is also a very favorable phrasing for a period of enormous human suffering. I think there is an arms race analogy to be drawn here, though the current situation and stakes are much different.
Funnily enough, there’s a good chance Truman never said those words; the earliest known attribution of the quote to him is from a Time magazine article written in 1988. It’s not in any of his personal writing or speeches.
'A man may do an immense deal of good, if he does not care who gets the credit for it.'
"QI hypothesizes that a Jesuit Priest named Father Strickland deserves primary credit for this family of sayings. This claim is based on the diary entry dated September 21, 1863 written by Sir Mountstuart E. Grant Duff and multiple attributions thereafter. However, similar maxims may have been developed independently."
For example, one generally don’t name cells in their body, and cells works generally pretty well to sustain a whole self.
On societal level, obviously no one is aware of who authored the biggest mass of benefit that is granted by the mere fact of living in society. No specific individual have credits for example regarding the language they use daily¹. For the most recent ones, it’s now often possible to attach some name to invention of some cultural/technological artifact, but of course Zeitgeist-wise perspective tend to consider it’s rarely a single punctual individual that should be considered the critical pivot point.
On the other hand, even Marx that was so critical of individual alienation couldn’t refrain to make the intellectual equivalent of pissing territory and attach a name referring to its own ego within all of his works. Ego is a masterpiece of illusion: it will bring the foundational leveraging to build every possible societal hegemony under the sun while convincing the deluded that serving the ego is the indispensable first step of everything to best serve every individual selfs.
¹ Even for constructed languages, such as Esperanto that I practice myself, whoever construct them obviously don’t start from cultural blank slate.
I don't understand. The AI-math guidance counsel has recently explicitly argued against what you say in paragraph two, and which is why for the Kakeya problem there was no group formed. The different communities have given different suggestions. I'm not sure how much I agree with the first paragraph either given these facts.
> We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies
1. They identified relevant experts for each of the discoveries they wanted to release
2. They collaborated with them to determine how to release the results
The effect for the first two (maths, comp sci) was the same because the experts they chose made that decision.
The downside here is actually the third case (physics). Since they only chose 5 scientists (David Gross, Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov) and those scientists chose to keep the results private until they finish their own work. Everyone else who might be interested has to wait.
That was my favouritism aside. Scientists working on similar or adjacent problems that are not included in these kinds of private releases might feel they are not getting a fair chance.
This is a process that will change, but it seems they listened to the anger from the math community and made changes that help address the concerns.
I think, sadly, these fields will collapse further into a tighter in-group. Anyone outside that group will find it increasingly hard to penetrate. Talent won’t get you there anymore.
From my experience in research this was already the case 15 years ago (and from older peers, way before). Science has always has "in-groups". I forget the name of some philosopher in the 50's who was the opinion science is just a bunch of fads and it takes the old generation to die before a new idea can really be accepted.
“A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” Max Planck 1949
Commonly paraphrased to: The knowledge quest proceeds funeral by funeral.
The hubris in the framing is astonishing, amusing and sad all at the same time. Many say, that's what HN attracts. Anyone who talks about inevitability except of death in these times is just someone who considers they'd be in the "have" rather the "have not" category.
I still can't see what this means for human understanding and how open would this knowledge be. Who'd ultimately get access to this stuff going forward and what kind of power games would ensue hereon.
I'm not sure why you think that someone who believes that this is inevitable is just someone who assumes they'd be in the "have" rather the "have not" category.
It seems like you are doing exactly what you accuse the parent commenter of doing.
The only difference, perhaps, is that the commenter seems to extrapolate past progress to future progress, while your own assumption is a lot more tenuous.
I just find it uncertain what lies ahead. I just don't see the sustainability in this. I have degrees in Mechanical Engineering, Applied Mathematics and Scientific Computing. If the "inevitable" is that I do manual labour till we can augment the robots to do that as well, I don't see how me or millions others similar to me have any agency.
Right now, a bunch of stuff is blurted out every month or quarter and people in notable positions are given a chance to "tidy it up". Let's assume that you have even the open weight models at your disposal, the hardware to run any of this in the kind of scale needed, requires capital of Governmental level funding.
Where I see this heading is that there'd be a divide of who controls the resources to produce and who gets delegated to do the menial work. If that's what you mean by my tenous assumption, please do share what you see differently.
Your assumption was that someone who believes that this is inevitable is just someone who assumes they'd be in the "have" rather the "have not" category.
I agree there may be some correlation, but it is less of a sure thing than thinking AI replacing mathematicians is inevitable, in my opinion.
It’s always weird when people are so quick to use a single positive thing to trivialize the many historically questionable things an entity has done up till this point. It certainly doesn’t come from a place of rationality.
What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
> What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
This is like going to a politician's event and seeing him giving away lollipops to toddlers and kissing babies. But it's the same politician caught in corruption and embezzlement scandals.
You look at that and praise how they have redeemed. Other people look at that and see a corrupt politician handing out candy.
People tell us who they are through their actions. If you prove yourself to be capable of morally reprehensible things to the point throughout the years you build a reputation for all the morally reprehensible things you do, does your potential to do morally reprehensible things change if you go way out of your way to let people know that sometimes something's you do aren't morally reprehensible?
Wrong, these companies have shown time and time again they are irredeemably misaligned with our interests.
Right now you and everyone else is doing the "leave the multi billion dollar company alone" version of the neckbeard "leave the lady be" meme. And you deserved to be chastised for it.
Its the equivalent of you going around telling everyone in the office about how your boss is turning it around because they bought donuts today, after they spent three years telling you you're worthless, they're entitled to taking all the credit for your work and they plan on firing you as soon as they can.
Anyone can solve any problem they want and release it. Nobody is defending multibillion companies, they are pointing out hypocrisy when the same would be celebrated of it was some academic mathematician who also used ai released a solution vs open ai or any other entity doing so.
This is going to be a very big deal in some of my circles, which include KITP (the Kavli Institute of Theoretical Physics), where David Gross was the Director for many years and now holds the Chancellor's Chair Professor in Theoretical Physics.
I’m not sure what meaningfully changed except for the QCD / yang mills result (where leaders in the field are going to do a review and write up first).
After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
> After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
This.
The truth if the matter is that this is still a publicity stunt. This investment of time, effort, and compute is aimed at building a reputation that this company's model is better than the competition's. The previous publicity stunt brew up in their face, and now they seem to be trying to adapt and double down.
I would have found it a lot more impressive if they just empowered scientists through some kind of grant rather than to internalize all this. It sets the tone for what's to come and inevitably if what they find is juicy enough it will not be published. For instance once they start working on materials science (if they aren't already).
Ok my original comment was snarky (apologies to all three mods), but you've raised one of my favorite topics so I can't resist engaging a bit more seriously! In the second clause, that is -- the first is trivially true in some senses and trivially false in others, so it doesn't seem very productive to nail down.
So Chomsky thought of himself as a philosopher of science more than anything, and his life mission is/was to answer a fascinating question: what kind of creatures are we? There's been a growing consensus going back to at least Darwin that we're separated from the rest of Animalia by our "reason" (!= intuition != understanding != judgement != volition), but he helped popularize an old theory[1] that puts it kind of beautifully: human language is only possible in beings capable of producing an infinite set of meaningful outputs in response to a finite input.[2] And so far, that list includes only two definite members: Homo sapiens sapiens, and computer programs built upon large language models.[3]
So all that to say: yes, I think you're wrong; the programmatic ensembles that brought us today's breakthroughs are an "infinitely deep well". As long as we have questions to ask, they are not theoretically constrained from answering them.
Of course, this invites some obvious follow-ups on practicality; even the most impassioned ASI boomer wouldn't argue that the exact ensembles used here could answer every question in the universe. But A) that's possible because some infinities are bigger than others, and B) the exact same caveat applies to humans, whose cognitive capabilities are thoroughly determined by accidents of biology, culture, and episteme. The latter is the thrust of Chomsky's famous debate with Foucault, which is absolutely worth a watch for anyone who hasn't seen it (starting around ~18:00, AFAIR): https://youtu.be/3wfNl2L0Gf8?t=1080
---
TL;DR: I ultimately disagree on both points, yes. In the ways that matter, machines are no more constrained than animals.
[2]: Many animals are capable of communication, often quite complex and fascinating communication at that; even flora and fungi communicate! But language is qualitatively distinct. You may have heard this in the pithy-yet-mindbending sentiment "we've taught many apes to communicate with sign language, but none of them have ever asked a question."
[3]: It's no surprise in hindsight that trying to build better autocorrect got us to AGI; I find it unspeakably tragic that Chomsky never got the chance to see how thoroughly he's been vindicated, in this regard. He had his stroke mere months into the singularity, and had a quite negative view of the approach before then.
> human language is only possible in beings capable of producing an infinite set of meaningful outputs in response to a finite input.
> And so far, that list includes only two definite members: Homo sapiens sapiens, and computer programs built upon large language models.
In LLMs that's a bounded infinity though, the models themselves have only so many bits and the activations are also limited. Once you reach the same state twice the model is looping and it won't break out of that loop. I've seen such loops across tens of thousands of tokens so I'm not sure to what degree the infinite output rule holds true. It would be an interesting experiment to try to figure out to what extent the 'busy beaver' can be kept busy before it degenerates to looping or nonsense.
LLMs are just regurgitating an output according to a set of rules generated from harvesting all human output.
Language to humans is a symbolic mapping of labels to reality based on actual experience.
"Language" to LLMs is just a string of tokens to manipulate based on relationships harvested from recorded human experience.
It doesn't mean anything other than 1s and 0s - it's not a language because the LLM isn't communicating with you, just as the sum of humanity isn't communicating with you.
It's just a really good "choose your own adventure" book.
...do you think molecules and their interactions are also not bounded physical systems? What part of a human being do you think isn't a turing computer? Please release your groundbreaking physics paper proving this.
They don't necessarily deserve credit, it is just the bare minimum courteous behaviour in academia to reach out to other researchers in the same area before publishing a result.
I'm also unaware of the quality of the consultation with the relevant mathematicians, colour me skeptical there.
> "The external collaborations will follow four principles... until Anthropic, Google, DeepSeek, Google and dozens of Chinese companies all make the same offering, in which case we will rediscover and re-implement all of the things you hate about traditional academic advancement and publishing, except because our companies are run by people even more mercurial and self-absorbed than university presidents, we'll wind up supporting even fewer people to do it."
ETH is false! I really would not have believed it.
[0]:
> We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula. For every fixed k ⩾ 2, on a formula with n variables and m clauses, our algorithm uses $poly_k (n, m)$ space and runs in time $(n + m)^{O_k(1)} 2^{n/(\log n)^{1/2 + o(1)}}$
The dominant term there is (roughly) $2^{n / (\log n)^{0.5}}$, so still superpolynomial but less than exponential. Fucking wild.
And the claimed algorithm solves the counting problem, i.e. determining the number of different solutions, which in many cases is much harder than just determining whether a solution exists or finding one of the solutions.
Yeah, this really blew me away. I took a graduate computational complexity course and the original prof had to drop out due to health reasons. We got another professor from a completely different field who was basically as green in the field as we were, but with far better research chops. It turned the course into kind of a long-running lab where we just kind of did whatever we wanted. We spent a lot of time just philosophizing (mostly uselessly) about the ETH. I am very chuffed to see this result.
It's not AI that has created those open problems. It has simply shown that some conjecture is false and that leads naturally to problems in a sense already posed.
A very different thing is proposing a framework and in it identifying interesting questions. So far AI has only worked within existing frameworks addressing problems already posed. That may change in the future, but it would be a genuine and qualitative evolution.
ETH is false! This is one of the big long-standing hypotheses that I actually remember from university, and afaik, it was reasonable to assume it would be true; reasonable enough for a lot of papers and results to assume this for bounds or hardness proofs of common problems in parameterized complexity, anyways.
Also glad to see that one of the authors of the paper stating it is also in the loop here.
What were the reasons to assume it would be true? I mean if you assume that NP < EXPTIME (which seems very likely) then you end up with algorithms that run in exponential time, but can't simulate turing machines with polynomial overhead. Obviously you can't simulate TMs with just anything, but most of the time... computation finds a way?
What will happen to fine grained complexity if ETH is false? So there will be a solution for the Clique problem? So our social networks will evolve differently after this?
Interesting that the physics result is not released yet. From the limited wordings, “unified completion of the Standard Model of particle physics” is probably SU(5), which is experimentally disproved as one describing our universe. So the wording sounds a bit confusing, ie they are not proving or discovering Grand Unified Theory (GUT), and there’s no Higgs. So it is not really the standard model, but models that’s closely related to that. It is still an impressive result. IMO at the level of the recent quasi Riemann hypothesis level result, shy of getting the millennium prize (no mass gap) but is actually a very meaningful result, both of which arguably more impactful that the one they solved a millennium prize problem (Navier–Stokes).
Capital allocation to advance mathematics and sciences is undoubtedly a good thing. But this does beget a few questions:
1. How much did it cost?
2. Would an equivalent allocation to incentivize human effort have borne the same results?
3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
#2 is particularly important to answer, given research grants have been getting cut elsewhere: see U.C.L.A's research grant cuts from last year and Terrence Tao's attempt to raise funds (about $60k) by collaborating with Open AI earlier this year [1].
As for #2, personally I feel like OpenAI's results are superhuman at this point. I don't think there's enough human talent alive right now to produce these sorts of results within a few years, no matter the budget.
There are lots of reasons I believe this. One is that frontier AIs are very clearly superhuman on shorter problems (as can be seen on benchmarks like FrontierMath, or just by playing with them yourself if you have a source for difficult problems)--so these achievements are "just" scaling that advantage to research-sized problems. Second, AI is also clearly superhuman in its knowledge across fields, so it can make connections and combine techniques that humans won't be able to access. Third, these problems are famous and have withstood efforts of many of the best humans, in many cases being considered unapproachable.
Altogether it seems like AI is not only smarter but also more cost efficient that humans for math-adjacemt problem solving. I'm not saying that AI is superior at everything in math, eg theory building, but I think it's really unlikely you could get any group of humans that could compete with OpenAI's latest results.
It depends on the definition of superhuman. For example my car is superhuman as it allows me to get in one hour to a place where it would take me ten hours to go by walking, and can I walk ten hours non stop? However it is not superhuman enough to carry me where I could not walk too. And there are places where I can walk to but I cannot reach by car. Maybe that applies to AIs as well. A plane is in the same category of superhumanity of cars, with more speed. But I could still get to Australia by walking and rowing with my hands on a log (and probably die, I know.)
However there are truly superhuman vehicles, spaceships and submarines, that can carry humans where no human could go alive.
I don't know if these AIs are superhuman in the way of planes or in the way of spaceships. I also don't know if they can do everything we can do by talking or if they have some limits.
I mean, a plane can fly but not reach a branch and stay still on it. A small drone, almost. My coding agents sometimes can't do the obvious but is it because of some intrinsic limitation or because my instructions are bad? So far my experience is that they are superhuman like cars or trains, not yet as planes.
>2. Would an equivalent allocation to incentivize human effort have borne the same results?
Unlike their initial massive brute-force run that solved the Navier–Stokes Millennium Prize Problem, most of their subsequent breakthrough results in mathematics have come from a single prompt to a single AI agent, using approximately 3 hours of ChatGPT Pro-equivalent thinking compute per result on average.
I think you are interpreting 2 as offering some large amount of money as a prize for discovering a solution. But I think it was meant more as funding a team of mathematicians to work on the problem.
> 3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
I think the scarier alternative may be OpenAI forever gating their super-frontier models, capturing a significant portion of the sciences and leveraging it for equity in businesses that build on scientific advances. e.g. "Hey CATL, how about you guve us 5% equity in exchange for a battery chemistry breakthrough our AI has found?".
It's roughly equivalent to the playbook Palantir had for the public sector when they marketed themselves as a company that had cracked data science. At least that's what they've been pitching back then. I can't find a lot on public disclosed deals based on that.
3. no, but models will be better, it'll be cheaper to do
It's not a snark. The scale of VC philanthropy is unprecedented and it fuels itself. All of it will eventually, one way or another, become open. Long story short – there is nothing to complain about.
In 2025, Hong Wang won the Fields Medal for proving the Kakeya conjecture in three dimensions. Here, OpenAI is claiming to have solved a stronger problem that proves the Kakeya conjecture in all dimensions!
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
To be honest this doesn't feel as such impressive as the others. It was recently proven in 3 dimensions, so it could be "just" an extension of that. I haven't read the paper anyways so this is pure spectulation.
The question for some is, is whether the original authors would have produced their work without building off from the work of other mathematicians.
(which is somewhat valid).
Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
IMO that there is any work at all to build from is the result of a long tradition of taking the next step. Professor to student and so on. The danger of AI here is we get to take a leap, say, 10 steps - but then there is no more tradition to have the fortitude and mentoring necessary to continue the progress.
It all gets derailed while betting on RSI and never having to do math research again (but if an advanced theorem is proven, and no one alive is competent enough to comprehend it, does it even matter that we get to call it solved?)
This is why the AI companies are making money off selling tokens and raking in investments for as long as it will work and is not looking to create the value themselves and making money off it...
They know that value creation is a dead end sooner or later..
If we suddenly skipped 10 generations of mathematical lineage because of AI, wouldn’t it also be smart enough that you could just prompt it to connect ideas from graduate level maths all the way to the proof!
I think the question is rather, what will happen to human beings when any possible personal intellectual accomplishment will no longer be seen as interesting or relevant by other human beings when AI can do most of it?
It's all well and good to say that we should just go on these journeys for our own enrichment, but even so they will just be doing it alone...or else not do it at all because they need a shared human effort surrounding them to do it, and therefore end up depressed and immersed in drugs or media all day.
Of course we're awed at what AI can do now, because all of us observing it have had the luxury of having such opportunities before AI. I got my PhD before AI and I found satisfaction in participating in a shared math community, which I wouldn't have done if I were starting my PhD now, because I would find the current academic environment unacceptable and useless.
Many people still do woodworking for the love of the craft, even though factories exist with machines that can do everything hand tools can but 10x better and faster. If AI gets way out in front of anything humans understand, I expect that mathematicians will probably pick up where their understanding actually is, and carry on.
That analogy is rather poor. Woodworking produces discrete, unique, and physical objects, that can be made, and found use for, as many times as the craftsman wishes. Once you discover a law of physics, you can't discover it again, and it's not yours to do with as you please.
> Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
I don't think so, it was a big leap forward in terms of creativity, imagination, rigor and experimentation. Einstein was able to "run" visual thought experiments inside his mind
That would be a very exciting experiment. I doubt you could find enough vintage training data to train a LLM enough to do it without it being contaminated by anachronistic data in the weights, but I would be absolutely amazed if it could reproduce that.
whether the original authors would have produced their work without building off from the work of other mathematicians.
That's rhetorical, right? Of course mathematicians built on what older mathematicians built and OpenAI's machine built on that. The real question whether OpenAI's machine has contributed something others will in turn build on.
Not saying I agree with the view, but something along those lines popped up when the Naive Stokes conjecture was seemingly solved (and way before when Open AI began solving Olympiad level problems and so on with their older class of models). Personally, I do see both sides of the argument. Though the broader impact remains to be seen.
I mean, the models (starting with Bel clearly and Astra probably--though Astra probably is worse than top humans in some areas) are superhuman at math.
I understand that the Exponential Time Hypothesis being false doesn't necessarily mean P=NP, but for people who are a lot more knowledgeable about complexity theory, does this update your priors significantly in the direction of P=NP?
It's been a long time since I seriously thought about complexity theory, but I'd say no. For similar reasons to why complexity class E is not closed under polynomial-time reductions.
The most important component in the time bound is of the form 2^{n / polylog(n)}, which is subexponential. But if you have reduction from another problem that increases instance size quadratically, the algorithm for the other problem is no longer subexponential. Which means we are still essentially in the same neighborhood as before.
It's a huge result if true, and probably the first major AI result that affects the papers I read. But since the boundary in the hypothesis was kind of arbitrary, breaching it should not significantly change our intuition on complexity classes.
I think the bigger issue (for someone hoping for P=NP) is that there's a huge gap between "polynomial" and "exponential." Even if it were the case that all NP problem classes had subexponential decision algorithms, that would not get us close to P=NP.
How much effort has been expended on optimizing those reductions though? I have a harder time believing that one can hang one's belief in P!=NP on that particular hook.
The challenge is the initial reduction from an arbitrary problem in NP to the problem you want to prove NP-complete.
In the usual approach, you assume a nondeterministic Turing machine (or equivalent) that solves the problem and a time bound (which can be an arbitrarily large polynomial). You then create a boolean expression that can be satisfied if and only if the Turing machine accepts the original problem instance. Unfortunately all constructions so far have included the time bound as a factor of the size of the expression.
I wonder if there are proofs regarding what neural networks like the transformer can solve. Because fundamentally, Astra is just a bunch of matrix multiplications and a few well defined non-linear functions.
Anyways, most of the times, SAT can be solved quickly, but not always, so getting the answer from Astra in polynomial time doesn't prove much unless you can also prove that it always does, no matter the input.
Iirc, there's a big difference between the computational complexity of what a transformer can solve and what it can solve in practice; what it can solve in practice tends to come from things like singular learning theory (which describes the forms of generalizations that a given ML algorithm can make), but I think (iirc) that transformers can compute a pretty broad class of O(N^2) functions?
The context can change each time. To implement a UTM only requires 2 states and 3 symbols or the reverse.
So if you use the context not as a tape in itself, but as an io channel to read/write an infinite tape, the context for any given step can remain fixed in size, and so the step itself can be O(1).
Of course it is. You’re conflating a turing machine with a transformer. Transformers don’t have a local read/write head. The [self] attention is over the whole context, which makes each step O(N) at best (with caching) and O(N^2) w/o.
Imagine you're trying to prove that water is wet (P!=NP). You lack a formalism that could make sense at building a proof even though pretty much everyone agrees from empirical evidence that the water is indeed wet. A promising attempt at building one such formalism was just demonstrated by a model to be a dead end.
I like this, but the actual finding is a bit different. Trying to keep the water is wet makes this awkward, but I believe it's more like:
Everyone agrees that water is wet (P!=NP), but no one can prove it. In the meantime, a whole field of computer science was built on a stronger belief (ETH): that water, at its worst, soaks. Turns out, even at its worst, water only makes things damp. It's still wet. That said, this finding did produce a new measurement of wetness.
> Anyone credible [who believes P=NP with some reasonable probability?
Do you accept
- Moshe Y. Vardi [1]
- Anil Nerode [2]
- Donald Knuth [3]
to be credible? Source for the statements that they consider P=NP to be possible: [4]
And you may include me in the list of credible P=NP believers because treat myself as a credible authority, i.e. when I ask myself whether something is right, I consider myself to be much more credible than some famous scientist. ;-)
(to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems. One should consider the evidence by oneself instead of believing in P!=NP because lots of computer scientists claim all the time that this is likely true, or consider P=NP now to be possible just because some famous computer scientists consider it to be possible)
Note that, while Knuth believes that P = NP, he doesn't believe that it will have practical applications because the exponents and/or constants will be galactic.
> to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems.
No, it's an irrational rejection of the notion of expertise. The experts are not "heroes", they are simply people who have done vastly more study than folks like you or me. e.g., if Magnus Carlsen looks at a complex chess position and says that white is winning, I will trust his judgment far more than my own.
> The main argument in favor of P ≠ NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration. — Moshe Y. Vardi
That sounds particularly relevant in how AI-assisted/driven mathematical research is changing the game, where previous assumptions based on human experience/effort are being overturned unexpectedly.
No, it is famously a pointless internet debate with no settled answer. There is no scientific definition of what can or can't be wet, saying that wetness only applies to surfaces is just an opinion. It's an entirely linguistic question, and most people would say water is wet. If you want more than descriptivism, the common definition is something like 'containing or being covered with liquid', and I think most people would agree that a mass of water contains water.
Water is not wet. Human lacks water sensor. All the sensors we have a temperature ones so we sense "wet" as a combination of temperature and visual information. Close your eyes, touch water of your skin temperature and you won't sense it. It's not wet.
I think it would have to if you're honest with yourself. Any proof of a novel implication of P=NP, such as !ETH, should increase your belief in the two sets' equality. Just as any novel implication of P≠NP should adjust your belief the other way.
My thesis was around showing implications of few different hardness conjectures. So grounded was ETH that I introduced it in a side note as Conjecture 0 after Conjectures 1 and 2. I don't think it's easy to express how substantial of a Bayesian update this is all around.
No. Scott Aaronson said that it changed it from 3% to 5% for him, which is a lot more generous than I would be (but he knows far more about the subject than I do).
> does this update your priors significantly in the direction of P=NP?
My answer is no ... and Aaronson's answer would also be no ... clearly he doesn't view a change from 3% to 5% to be significant ... his response downplayed the significance of !ETH in re NP?=P. So apparently you downvoted me for simply stating my own opinion while you completely misunderstood what it was an opinion of.
For me, this result is significant. It basically adds much more hope P is equal NP (that majority seems being against). I believe one day AI will explain a major flaw in modern mathematics, so a kind of correction in axioms will be suggested to prove Rieman's hypothesis (or a completely new apparatus)
That's not that how that works btw. The incompleteness is a property of the algorithm which does the proofs, not the theorems. If a theorem is true it has a proof and a program which proves it (printf("the proof verbatim")). But no computer program can successfully prove all theorems. The space of potential proofs is too big to search systematically, but you can still get lucky.
That is also not how it works. The theorem says that there exist some true statements for which there is no proof. This is assuming the axioms are consistent.
Can something be both true and unprovable without being a self-referential Godel sentence? I assume most statements that aren't self-referential are independent, like the continuum hypothesis with ZFC.
Either P=NP or not. Hoping one way or the other changes nothing. Nor does it change how efficient computing is, nor does it have any effect on cryptography. Discovering a vastly more efficient means of factoring primes--perhaps via quantum computing--would do that. But just learning that P=NP does nothing to further such a discovery. Almost certainly, if P=NP, the exponents and/or constants in a polynomial factoring algorithm would be galactic. (FWIW, Knuth believes that P=NP and that it has no practical consequences.)
> Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026, which led us to develop the collaboration described here
Yes, people did. Well at least I did. We knew we can use QCD to compute loops and to some degree masses in the lattice. But there were serious doubts the underlying mathematical theory was sound.
For many it didn't even matter. It would just mean that at those energies the theory is not even valid.
I am not certain on the extend of this proof but QED is probably not sound on it's own and we have been using that even longer. lambda phi^4 behind symmetry breaking and the higgs is strongly believed to be unsound on its own.
Yes, but it has also been known for 50+ years that QED has a Landau pole and might be UV trivial or inconsistent, whereas QCD has asymptotic freedom and is thus controlled at UV.
More to the point, QCD has been experimentally tested both at UV (deep inelastic scattering) and at IR, while the physical meaning of testing QED at UV Landau pole regime is unclear.
I find it a bit ridiculous that they claim three discoveries in "math, CS, physics" when they are basically all pure math (in particular, they all may be formalized in Lean) and only belong to these different disciplines for historical reasons.
Think openai should start including visual and interactive explainers in these where the viewer can toggle level to get a more accessible/technical explanation as needed. Would also be a great demo, and perhaps increasingly essential for us to know that explainability is not lagging behind provability.
I think at this point openAI does not really understand the results themself and any easy to understand visualisation would imply understanding. I would rather wait for the work group to produce some explainers or at least approve visualisation
Lots of progress recently in formally verifiable domains. Really curious if this is a product of the machine learning folks behind most large language models being much more likely to have a background in math, physics, etc. Or if it is that large language models have some fundamental limitations in the more "scruffy" sciences like biology.
My guess is that in formally verifiable domains you can use RL-esque processes to get lots of good high-quality training data with reward signals (does it verify?), but the same does not apply to "scruffier" work. I would also imagine that the data volume of high-quality scientific work in "scruffy" domains is too low for the large language models to have a good manifold for "high-quality biology research". Would love to be proven wrong if anyone knows of anyone doing great scientific work with large language models in non-verifiable domains though.
Many of the harnesses used to search for solutions rely on quick/cheap correctness checks to discard incorrect hypotheses. With something like biology, you might need to run a wet lab experiment to reject an idea. This can add days per iteration, slowing progress down by a factor of 100 or more.
The sad reality is that many computational approximations for biological phenomenon are relatively weak and lack the predictive power we need.
Is there a path where compute gets poured into developing, and then eventually using, computational models that supersede the shortcoming of the existing computational approximations you mention, which thus makes computational analysis and developments possible at rapid scale?
In an area I am more familiar with though still not my domain: when the stiffness matrix and computational linear algebra methods collide, it unlocks a world of computational structural design and analysis by commoditizing the underlying mathematics. This is a simple example, but you get the idea.
Obviously with CFD, mechanics, etc there is always massive work being done to try to unlock new computational methods, faster/parallelizable computational methods, differentiable ones, etc etc, but the general feeling is that the core physics are well modeled (NS millennium problem not withstanding) and are not lacking in predictive power for useful applications. I would have assumed that the same would be generally true for computational biology etc, so just curious to hear you elaborate more. It seems plausible to me that turning the brute force of “the models” (as in, swarms of AI compute) to work on/develop/solve/address whatever representational problems there are in the computational approximations/numerical methods you mention, particularly if that is a bottleneck to performing more extensive methodological/experimental exploration in silico to enable better exploitation/use of limited resources in wet lab.
Again, I know nothing about computational biology, just stating my intuition and curious what your take is. Probably the limits of my knowledge can be demonstrated by the fact that the only modeling methods that come to mind as being vaguely relevant would be, like,… molecular dynamics, maybe discrete element method in a weird way but doubtful, and various pop science understandings of protein folding etc.
Computational biochemistry is what I'm currently pursuing, but I'm pretty new to the field so take everything I say with a grain of salt: Biochemistry is a really really really tricky problem, for a couple reasons:
1. Biology has a very nasty habit of not cleanly abstracting. It's not like we can treat every protein as a "diffuse mass" without losing a ton of important interactions. This is unlike something like CFD where you can treat a mass of water molecules with certain constants. Think of something like cancer: a small mutation to a DNA strand can cascade into a tumor. You have to deal with the mutation happening at an atomic level while scaling to a whole organism (yes I know there's models that can approximate this, but they can't capture all the interactions).
2. This brings us to modelling at the atomic level. To model a system of atoms with the Schrödinger equation, it naively scales O(6^n) per particle. Note that means every nucleus and electron, so if you have say 10 carbon atoms and 22 hydrogen you're already at 10^127 operations. We can cut corners a bit and end up with something like CCSD which scales O(n^7), or cut more corners and scale O(n^3) for DFT. DFT is pretty much the last method where bond forming and breaking is modelled, so after more corner cutting you lose losing reactions, which are important for understanding why something is happening. Most protein folding simulations (moreso prior to alphafold) run a very crude approximation that models atoms as balls on springs (things like AMBER and CHARMM), which obviously can't handle reactions.
3. If this wasn't bad enough, you also can only move through time at about 1 femtosecond (10^-15), so it takes a ton of simulation steps to get even a nanosecond in. Forget millisecond or second lengths where things start to happen.
So, while the math of the physics exists, computationally it's practically intractible. There's some exciting work with something called Machine Learning Interatomic Potentials (MLIP) which approximate DFT using a neural network which are about 1000x faster than DFT, and iirc scale much more linearly (look at things like MACE and orbmol if you're curious). So we're making progress, but we've gone from so ridiculously impossible to pretty much impossible for large systems.
I have a couple ideas about developing custom microchips to make this faster, which is why I'm pursuing this, but there's still a massive way to go!
To add to this I think there are “reductionist sciences” and “systems sciences”. Reductionist sciences are ones, like physics where there are lots of copies of a thing with near identical properties at different scales (the periodic table would be considered an incredibly short list and that seems to be one of the longest list of “different things” in physics).
Systems sciences are ones where each scale has lots of things with unique properties. If you think of “biological things between 2-4m” the amount of unique objects with unique properties and unique interactions is absolutely mind boggling. These sciences generally do not lend themselves to clean highly parsimonious models. The best you can do is coarse-grain a thing as a system (some very roughly defined set of interactions between different things) with certain behavior while accepting that you are losing enormous amounts of predictive power.
I suspect AI will remain worse at systems sciences for a long time because the “possible space” of models for a particular system is much larger than in the reductionist sciences and navigating it takes a human taste and instinct that LLMs don’t seem to have (yet).
One other thought, I think another aspect of biological sciences is just how much effort it takes to do research. Hypotheses are a dime a dozen, but actually verifying these is quite difficult. It takes a lot of patience, time, attention to details, and even then it often fails to replicate.
I don't think LLMs will just magically be able to connect all the dots, because "garbage in, garbage out" still applies. And there's just too many invisible intermediate states, there is literally just not enough information to fully constrain a model. Hence my interest in scaling computational chemistry to get these intermediates :)
Heh. Unfortunately computer science is really the weird one out where open source is the norm. A lot of computational chemistry knowledge is mainly transferred in research labs and through college classes, and you have to pay for a lot of the tools. But hopefully I can at least point you in some useful directions. Here's a bit of an overview of the CC workflow and free options (note I haven't tried all the free options):
1. The first thing you need to do is start with some atomic structure. You can often find pre-existing pieces (through websites like pubchem), but you need to cobble them together. I use chemcraft and gaussview for this, but those are paid options. I've heard good things about the FOSS program Avogadro though.
2. Once you've built the structure of interest, you want to calculate properties of it. Note that we only infrequently simulate the atom's movement, since that's quite expensive. There's a lot of techniques that let us get the most bang for buck, like calculating transition states, or states of lowest energy which we call intermediates, or running something called an IRC, which follows the molecule to its lowest energy point (note we pretty much always think about these things in terms of energy). The traditional programs I use for this are Gaussian and ORCA (ORCA is free but you do have to create an account on their forum). Newer methods include MACE and orbmol, which are pretty much only used for trajectories (e.g. running through time) as far as I know. This newer crop of programs are much more FOSS, but they aren't as rigorous so my lab uses them only as a "rough draft". I pulled in one of my current notebooks if you're curious to try it out: https://github.com/smj-edison/qc-tools/blob/main/examples/ml...
3. Interpreting results/debugging problems. This is currently done in my workflow using a python script to extract values from the job runfiles (oh yeah, did I mention the terrible DX? A lot of this stuff is done by hand-writing a job text file and then using regex to parse the logfile). We do a lot of analysis based on relative energies (this is something called transition state theory of reactions).
I definitely glossed over a lot of details, but this is long enough as is. Definitely let me know if you have questions! If you do want a "hello world" though, I'd definitely check out the notebook I linked to. You'll have to create a file called "start.xyz" with some sort of chemistry program (or use AI to pull one for you), and then you can run the system through time. I would use a system I'm researching right now, but I'm not sure what our lab's policy is on broadcasting things to the world.
I suspect the line forward is generating improved predictive models and implementations of those. It’s something LLMs can aide with, although they’ll have slower progress than domains that do already have robust, cheap validation checks. That’s probably why those domains are being targeted (they are the paths of least resistance to successes, which is why they’re being pursued).
With that said part of the reason some domains may not have robust cheap verifiers is because they’re difficult to produce already, so it may be a bigger lift.
The bottleneck in biology isn't thinking, it's experimenting. AI could potentially operate a lab but the test process is a lot more expensive than math is.
Are there efforts into building "autonomous laboratories"? Not sure if the idea is practical but it seems like a direction things might go in, where you give an advanced model, or a swarm with a leader, free reign to run experiments day and night.
It's because LLMs provide an efficient way of optimizing a complex, non-linear search space when there's a way to measure progress. The less verifiable something becomes, the harder it is to optimize (but not impossible).
Super cool research! I think this is still a formally verifiable domain but different than the kind of formal verification I was thinking. Maybe if LLMs continue to be great at this human science turns into making “fuzzy domains” much more formally verifiable (although I don’t think this is possible in some areas).
Obviously it's easy to understand why having strong abilities in maths and coding can help with solving problems in physics. But it's also true of knowledge and mental models more broadly.
For example, say somehow you managed to train a frontier model without any data about elephants. If you did this it would be relatively easy to later train elephants into the weights. You'd basically just need to train it about elephants at a very superficial level. After all, it it already knows what a mammal is. It knows what legs are. It knows what eyes are. It understands how it's digestive system works. All the things that makes the biology of elephants difficult doesn't need to be learnt from scratch.
I assure you people on HN will be surprised at how quickly frontier models will become competent in domains that are harder to RL. They won't need anywhere near as much data as people currently believe.
>We give a deterministic algorithm for exactly counting the satisfying assignments of
an arbitrary k-CNF formula [in superpolynomial but subexponential time]
I assumed all new breakthroughs in advanced complexity theory would be theoretical and non-constructive, as in "we prove that this algorithm must exist but we do not provide it".
I'm genuinely shook at today's results, specially at the fact that there _is_ a constructive algorithm.
Polynomial is something like exp(n log x), where x is the problem size. Exponential is something like exp(x). Superpolynomial but subexponential is roughly when the exponent is between n log x and x. An exponent of (log x)^2 would do the trick.
Thurston wrote about this in “On Proof and Progress in Mathematics,” including how his own approach changed from proving theorems to helping a community understand the mathematics.
Nicolas Le Roux, a Lagrange Prize winner / prof / strong mathematician, turned me onto that around the time I finished my PhD in ML last year. I have been thinking about it a lot, lately, for obvious reasons.
At first, it felt silly to grapple with what science is only after finishing a PhD. This publication from OpenAI finally pushed me over the line to publishing an essay about this struggle between AI science and human science: https://news.ycombinator.com/item?id=50253759
This is so far above my head it isn't even funny. I just looked up ETH and was 9 links in before I realized even the explanations of the parts of it were out of my grasp. Can someone explain the implications of this?
It depends on your background. If you don't have a computer science background the statement needs layers and layers of buildup that can't happen in a single hacknews comment. But the outside implication is still accessible.
The outside implication for what is going on is that OpenAI has very clearly demonstrated insane capability in coming up with clever algorithms. You cannot deny some real level of super human like abstract reasoning here.
K-sat is the hard theoretical computer science problem. Nothing else is close in terms of how much it has been studied. There are journals and conferences for SAT.
It is really hard to come up with faster ways to solve it. So much so that people were convinced that you really can't go faster than exponential time. OpenAI has just shown a breakthrough construction better then decades of research by elite specialists was able to come up with.
They do that to make it easier for readers to verify the statement (which is entirely contained in that file) separately from the proof (which is spread over thousands of files and github can't even show them all).
I glance at the HN headlines but rarely have time to dig into the details (and the few times I read 30 mins or more, a lot goes over my head).
Are there any good accounts to follow (platform agnostic) that give occasional updates on the intersection of AI and math that aren't too dumbed down, i.e. that a layperson can appreciate?
Regarding Yang-Mills, as of a few weeks ago there has been some expectation that this (specifically completing Balaban's program for the existence part without the mass gap) could happen, though from the Anthropic side: https://nitter.meowing.monster/ElliotGlazer/status/211115394...
(Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)
I am not sure if I side with Tao on this, or 3B1B. It's already causing quite a kerfuffle (to say the least) in the academic communities. This is the new norm now.
These resolve 6 entries in my top 500 open problems list (LLM ranked, https://www.proofatlas.ai/open-problems/) and there is partial progress on #3 overall (Yang–Mills Existence and Mass Gap):
Well, this is certainly different, can't say they ignored the criticism. Wonder if this is a monkey paw curls moment; if the findings continue to accelerate, most of it just isn't public now.
I'll give you an example from my field (nuclear fusion). I have a small team working on an idea which I think would really advance the field. Our initial concept paper was published in 2019. Progress on it has been slow because we don't have enough manufacturing capabilities, budget, or time on the facility. Will AI solve that? Don't get me wrong, I'm pretty bullish on AI, and it's been phenomenally useful so far (I definitely feel like a 10x physicist in the last 6 months), but I don't see a swarm of agents putting fusion energy on the grid tomorrow.
Also, fwiw I have a personal test problem I give to every frontier model on release, which is to ask it to final a realatively obvious solution (in hindsight) to a minorly important open problem in my field. I know the answer, but havent gotten around to writing it up yet (it's really not that important). Every frontier model (so far) just regurgitates existing solutions.
How much effort do you put into prompting it? Are you asking it one shot with a simple prompt? Are you giving it to multiple agents and giving them a workspace to collaborate in? You might give it to a trusted colleague who does not know the solution along with an agreement to jointly publish, ask them to go through the process of solving it with agentic support, and then compare your solutions.
Edit: to be clear, I completely believe your experience - just pointing out that it might be interesting for you to actually “try” to solve it thoroughly with the tools, though it is obviously hard to do that in a fair way when you know the solution.
Enough! I think it's a reasonably fair test of the models. Let's put it this way, I use the same process to resolve problems outside my peak expertise, and I find it works well.
I think the solution is easy enough that any competent colleague would come up with it. Some variant on it has probably even been published (hence why I'm not chomping at the bit to publish). I think it comes down to the models being extremely spiky, and the problems they're trying to solve being easily verifiable in code.
With my personal little test problem, that never made a difference.
However current frontier models got good enough at websearch and thinking through the problem, that they can finally piece together the answer from my own public writing (eg on stackoverflow). And recently someone wrote up an answer to the problem and put it on Arxiv.
Iirc some of the prior math discoveries involved very very very heavy handed guidance to force the AI to discard prior assumptions, and to try many avenues with MAX effort. So ya models don’t behave like ground breaking theoreticians by default.
Ha, I had a similar little test problem, but the frontier models are now smart enough to read enough online of what I posted here and there to piece together the answer. And someone else has written the paper in the meantime and got it accepted for publishing.
Definitely. It will let some smooth-talker with a vibe-coded website raise billions for "Nuclear Fusion AI". Then they can use AI to squeeze your team with lawsuits until you agree to sell cheap, and maybe keep you on as an employee to build it.
Hah. Fusion scammers are a dime a dozen. Unlike SaaS apps or finance, though, you can't sweet talk a neutron into existence, so they fizzle out pretty fast. I'm more worried about fusion dying because nobody comes up with a good enough idea than someone stealing my ideas.
Going on a tangent: I think fusion is pretty much useless this century. We already have energy generation technology that has high capital costs and low running costs: it's called nuclear fission and works fairly well.
Fission hasn't exactly taken the world by storm.
There are a few nice advantages fusion has, but they are more important once we want to reach the stars as far as I can tell; not this century on this planet.
But I'm just an armchair analyst. You probably have a much better understanding.
Nah I agree with you. Not this century, not this planet. Fusion rockets are really efficient, though ;)
I just find it to be an interesting problem! My hope is it happens before I die. My rule of thumb is anyone who says energy on the grid in < 10 years is a blatant liar.
I mean, you never know. If we could have simulation models with excellent fidelity, then AI could do all the research and iterations and come up with something very plausible. Cutting down physical world experimental iterations/prototyping.
So effectively, you would spend the first 1000 experiments inside a computer simulation and move to the physical world only when you had enough confidence.
I think you should realize that we already do that. In fact, that's one of the reasons computers were invented in the first place! It turns out that science is about unknown unknowns, and by definition simulations can't answer that.
But agents are really useful for doing some of the really boring stuff, like convergence checks or setting up meshes or whatever.
If you're talking about things at the equivalent level of say, finding out if positrons maintain symmetry with electrons for their magnetic moment, then yeah - I hear you. Or if you wanted to research some unknown behaviour of hot plasma at 15 million C, you can't do that in a simulation.
..or can we?
Humour me for a moment. Maybe simulating to the degree that was necessary for cutting edge work was not possible earlier, when humans had to program each and every thing in the system that could affect the result. But with current gen petaflop-on-your-desk level compute + swarms of agents that can work tirelessly to account for every known thing to humanity, maybe the threshold has been pushed far enough to produce some more meaningful discovery work in simulation?
I've got frontier models doing pretty sophisticated nuclear reactor design and analysis using open-source nuclear fission reactor tools using open-source reactor designs as baselines. It is WILD how good they do.
Doing AI assisted enggineering to explore a space and then using classical methods to validate a point design for licensing is a good approach. In licensing the vendor and the regulator check things and then also verify many times during startup. The overall safety remains good.
Up front: I totally believe you are doing your level best so please don't take this as trying to take you down, I just want to understand. In the hypothetical case that in 10 years time it turns out that that design was flawed who will be on the hook? You, the AI vendor, the regulator, the device vendor?
The thing that I've not fully groked here (might be relevant) is that OAI is generating "paper" proofs first, and only then formalizes them in Lean as a separate step.
They've as far as I'm aware not said anywhere how many write ups failed formalization due to a bug in the proof. If we assume that number is low, models would be quite good at one-shotting math proofs. Then it depends if you can replicate the math RL training for chemistry/biology/medicine. Is there enough data out there for that? No idea.
I'm seeing estimates of a %95-91 failure rate on attempts in generally (or, conversely, a max of ~10% success, but likely lower, since we can only go based on how many attempts they are willing to mention in their statements). E.g., from analysis in https://medium.com/@mehmet.ozel2701/openai-tried-about-4-000...
I wonder how many of those founder at the "paper" to formalization step, or at earlier or later steps (e.g., proof doesn't turn out to prove what the paper said, hits a bug in the kernel, etc.).
based on how bad these models are at one-shotting complex real-world software engineering problems -- it always has to be a conversation -- there is absolutely no way they're one-shotting complex math proofs.
Also, goes without saying, but put Sam Altman in jail, obviously.
I would trade (as if I have any say lol) every single breakthrough they have made in math/CS for _any_ good progress towards a universal cancer cure (very theoretical, I know).
I'm kinda hoping they are investing just as much into that type of research. Its their money but man... I'd say 999/1000 people are unaffected by all these breakthroughs. They probably have no idea tons of legendary math/CS conjectures were proven true/false. If a breakthrough in cancer research were to be made, it would very quickly start effecting everyone on earth.
The time required to verify is much higher, so some combination of vastly speeding that up or vastly increasing the quality of individual "thought steps".
It’s accelerating. We can automate scientific discovery. Imagine the amount of knowledge we may have over the next several years. It could be staggering.
How are people getting these results? Are they correct?
I use these latest available public models daily in max effort modes, and they barely seem to work and need extreme guidance and as much context and constraints as possible.
I find that unless the outcome I'm looking for from the model is almost binary and easily verifiable, it is basically faster to do the work myself.
Assumptions made by the models multiply and by the non-deterministic nature (dbf, similar to work by humans) I'll have something end up light years away from where it should be. During a normal workflow, I have to contend with every line I write forcing a very tight feedback loop... I suppose I could also sit there and spend 9 hours writing out a "perfect spec" but we've already tried that with waterfall development!
So, not sure if its the models, rather the class of problems I'm applying them to... but it is a bit ironic they are allegedly able to solve these conjectures but not stop to ask about assumptions.
Chasing open questions, it seems to me, is worthwhile for the things learned along the way. Do you expect these results will be helpful, or is it just a sign that we need a new way to navigate uncharted spaces now that longstanding problems don't stand long enough for us to organize around?
It feels a bit like having "visited" a place because you had a layover there.
I don't understand this dialogue, which seems to repeat with every result of this sort. Of course this is interesting! And of course the answer tells us something! And of course we also want to understand where it came from! None of this is in contradiction!
Knowing the answer to Fermat’s Last Theorem wasn’t valuable. But the pursuit of it led to many developments in other fields of mathematics that were valuable (where value is measured in terms of $).
I disagree. We climb mountains for the sake of it, and the rest of the world benefits because the gear improves and makes adjacent, non-for-the-sake-of-it activities safer.
We chase open problems for the sake of it, and others sometimes support us in doing so because the paths we pave along the way end up being useful for them for adjacent purposes. If you get the answer for free, you might have scratched the itch, but nothing was built for others to use, no fodder was found for the next great question.
Nobody pays mathematicians for solving famous problems. Famous problems are how mathematicians orient themselves while they do the other things that they're paid for (maintaining a useful community, mostly). All this does is deprive them of a navigational tool.
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Your formulation is a little bit exaggerated, but I (and many, though far from all, programmers) would agree with the following more "tame" formulation:
Getting software to solve a business problem in an elegant and maintainable way means that we got the abstractions right. Similarly, if a software can easily and elegantly be modified to cope with new business problems, it is also a strong sign that we got the abstractions right. In this sense, the business problems that the software solves are indeed just waypoints on the journey of finding the proper elegant abstractions for putting the business problems into code.
Turning these novel insights about "good" abstractions into code and programming languages enables the team to iterate faster and generate business value faster. So, appreciating that it indeed all about the journey means a commitment towards an approach to accelerate the rate with which the team can turn business insights into code and code into business value.
I know that many managers do hate such a formulation, but I stand behind it.
Often a problem is important because we have no idea how to solve it, which means we need to develop new machinery to solve it. That new machinery can then be used to do things we couldn't do before. Stimulating the development of that machinery is the most important part of the problem.
You should remember that mathematicians and programmers working for a company have different jobs with very different goals.
This is such a short sighted take. Number theory was considered beautifully useless for centuries, and now it’s the backbone of the cryptography that secures every bank transaction. We don't know where the pursuit of knowledge will take us.
Thinking in such a utilitarian way feels like a flavor of Meditations on Moloch.[1] Just look at how short-term, quarter-to-quarter thinking is affecting the economy and the lived experience of every human on earth. It’s horrible.
>
This is such a short sighted take. Number theory was considered beautifully useless for centuries, and now it’s the backbone of the cryptography that secures every bank transaction
If you refer to RSA or Diffie-Hellman: the number theory that it is based on is from 17th to at most 19th century, i.e. it precedes modern number theory.
I do consider modern number theory to be fascinating and do claim that many results from it could have huge applications in society, but the argument that parts of modern asymmetric cryptography is based on number theory is rather an argument that we don't need any number theory research anymore because these insanely old results that are used in cryptography have already been known for centuries - so no new research is necessary. :-(
We didn't know what investigations were going to lead to RSA or Diffie-Hellman and any particular breakthrough is standing on a mountain of prior work that didn't have an obvious application but nonetheless was necessary to get to a breakthrough.
Honestly I can't believe I even need to state this. Its literally how science works..
Knowing the answer is valuable, but it's not everything. We also want to understand why. The four-color theorem has a computer proof, so we know it's true, but we learned much less from that than we do from a human-understandable proof.
Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.
If you were a math grad student, would you rather your advisor be telling you to LLM maxx your ideas or working with you and giving you ideas on your thesis personally? I know my answer.
To say that we must have people that know how to X, because otherwise no one will be able to teach how to X, is kind of circular reasoning. Why is X important to begin with? What’s its inherent value to society?
I could address your situation in several ways and it would be a long discussion that a quick soundbite wouldn’t really satisfy. But at the very least, your argument is one of the arguments I would use to question the privatization of pure math by LLM companies. Pure math has little value to society so AI companies should consider not getting involved when they aren’t really making life better for anyone and are possibly harming young academics.
But to give a brief answer to your question directly: In my view, and I think many if they really thought about it, the point of life is to overcome. If there is an AI that spits out the answer to questions that don’t directly impact human wellbeing instead of letting humans explore the answer at our own pace, we should question whether that AI is worth keeping around or at least relying on too much.
Obviously the question is more nuanced when it comes to math, because even pure math can have applications to understanding reality and improving human wellbeing.
So the question then becomes what do we do with the AI when it can both improve our wellbeing and remove meaning from our lives? Is a life without struggle worth a life without meaning? Personally I don’t think so, so I think we have to optimize all of the benefits of AI while essentially taking a harsh stance against all the harms. Currently the main harm is to education, and that’s why I think we should question and probably even disallow its use by students still working to demonstrate their skills.
This is very well motivated. It's a bit like letting a combine for knowledge loose on that which made life satisfying. Having the answer isn't necessarily going to move anything forward. I also think that OpenAI is making a strategic mistake here: they are competing with their customers. That's something businesses usually only do when they are ready to ditch those customers.
the assumption is that training llms is greater than training individuals. llms may lead to their own "breakthroughs" but so have many people without the use of llms. is the thinking that we'd be dependent on llms and so we are now forced to be only able to discover what llms find discoverable? sounds like a loss
> If knowing the answer is not valuable then asking the question wasn’t valuable to begin with.
Let's say it's possible to know an alien civilization exists 100 million light years away from their waste heat. But that's all we know and it's unlikely we'll ever interact with them.
Was the question of whether we're alone still "valuable"? Wouldn't a lot of philosophy and even some physics fail to meet your definition of valuable?
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Imagine the next version of Claude Code or Codex no longer hands you source code and instead turns a natural language prompt directly into a compiled binary that passes the test suite. Would this programmer's boss ship the binary straight to production and dispense with the company's git repo altogether? After all, the goal at the end of the day is shipped software that produces value, not human-readable source code.
This is an exciting time, but also there are very real changes at hand. Game changing discoveries will likely not happen at universities but at private AI frontier labs. Moreover, the models that are used on this are not always open to the broader community today.
What will it be like in two years? Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies? Still thinking about the ramifications ... amazing and discombobulating at the same time.
I disagree with that. Commercial entities are very different from Universities. They are sharing now, but they don't have to continue to share in the future, that all depends on what they find.
> Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies?
The 3 breakthroughs all have implications for cryptography and nuclear technology, right? Hard to imagine nation-states allowing private companies to publish these results willy-nilly for much longer.
I wonder why the physicists are the only group to not want the full information released upfront? As a casual fan of physics[0] it seems like their completion of the standard model is possibly the biggest deal of all these breakthroughs, am I wrong?
The writing is a bit ambiguous, but if I'm parsing it correctly it's only a big deal to the small subset of physicists who care about mathematical rigor: proof that Quantum Field Theory (i.e. what theorists have been using since the 1940s or so) is mathematically sound.
This part,
The existence proof pertains to the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model
As a slightly less but still very casual enjoyer too, I don't think there is a huge amount of value in dumping the lean proof or the natural language proof. I'd rather have an abstract of a paper that I can at least hope to understand 75% of before getting lost.
I'm guessing if you were really interested, you could email YANGMILLS4D@gmail.com (itself a hilarious e-mail address).
Careful, completion of the standard model ≠ theory of quantum gravity or theory of everything. The actual question is too abstruse for me (biophysicist).
The QCD and Yang-Mills results don't seem to go beyond what the experts know/expect. Basically, they're saying there is a path integral measure, but shrugging when asked what functions are integrable with respect to it. The latter question is key for mass gap and confinement, and existence of the measure only means that the question of whether a mass gap exists makes sense.
There's a large logical leap from "we have very useful raw research data" to "we should give this data only to a small group of people." They explain that "the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community" but we are left to wonder why the experts need to be empowered to foster anything. In the case of immediate release the experts will have their hands on the same data as the rest of us peasants, and then they can go right on fostering.
Not enough chatter about the QCD result here! I'm of the opinion that whatever the correct representation of the universe is, the GUT (Grand Unified Theory) will be authored not by humans but by LLMs.
I thought it was interesting that in the original drop of papers, they announced much more progress on most other Millenium problems than on Yang-Mills existence. At the time I thought this might mean that it was a more stubborn problem, but it turns out that they had made a lot progress after all.
The Yang-Mills Millenium Problem has two parts though. The second part is to show that Yang-Mills theories have a mass gap, meaning that the lightest particle has non-zero mass. This is important because experimentally we know that there is a mass gap, so the theory must have one too to in order to actually be the correct model of nature. I wonder if they again have unannounced progress here.
> In our view, these manuscripts are best understood as a form of raw research data.
Much better!
> For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.
> All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.
Very good!
I was highly critical of their approach with the earlier mathematical publications, but this is much improved.
Most importantly, they aren't releasing slop papers that would never have been published in any self-respecting journal or made it past a deep round of peer review questioning.
> We are committed to ensuring funding for researchers to independently examine, explain, and build on the results above.
> We continue to invite feedback on the best way to disclose results and engage with scientists to accelerate discovery.
Well done!
Now you need to actually LISTEN to the community and the feedback, so you aren't done yet, but this is a much better start than last time. Keep it up!
Woah, they got QCD! I didn't think they could do that. Three things jump to my mind:
1. Is it really true? Did they get the basic conditions right?
2. Would I ever be able to understand the proof
3. But the most important one. Can we compute hadronic masses now?
This is an incredible scientific achievement. One of our most fundamental theories is mathematically sound. Without the use of AI is very likely I would have never seen this in my lifetime
I'm still shelled shocked, but my initial reaction is caution. Wait what did they claim to have proof exactly?
I can't see their pdf in natural language either.
I wonder how much compute they are throwing at problems in ML like continual learning and sample efficiency. Also how successful they are with these. Results like in the post make me hopeful we'll see some type of so singularity soon
It uses sparseness. I noticed a simple thing way back that for a n-boolean variables, the average size of a satisfiable boolean expression must be 2^n bits - basically every binary number that satisfies it listed out. So if a boolean expression of 4 variables is satisfied by 1111, 1110, and 0000, the number denoting it is 1100000000000001, and for every such number there is a boolean expression of 4 variables and for every expression one number of 16 digits. So it’s a fool’s errand to make a universal solver when average inputs scale at 2^n, if a solution exists it has to leverage the small size of the problems we actually care about, sparsity. Based on the AI description that’s what they did, something with polynomials, but it’s too soon. Looking forward to finding out.
Yes. Up until now, it was unknown whether containment of NP in EXPTIME was strict or not. (Obviously you can brute-force all possible witnesses, so any problem in NP can be solved in EXPTIME, it was open whether you could do better than that.)
Showing 3SAT (technically #3SAT) can be solved in less than EXPTIME resolves this. It's only slightly subexponential, not even quasipolynomial, but it is a very surprising result. And that too a constructive algorithm.
this is the first time one of the big labs has published anything related to my field. and i'm actually pretty excited? im more on the applications side of QFD than the theory side of things, but gosh i actually cant wait for the results to be released to see if i can use it for anything! i kinda hope more results come out, in more ways relevant to what i do, its so cool to see any sort of progress! wow!
I'm a little surprised the article stresses Lean programs so much, without ever stating what that is or why it somehow proves something. Maybe that's expected knowledge for this sort of thing.
It is because people take lean proofs at face value and you don't have to actually explain it. I have some math problems I am working on and while I had code showing it does the right stuff, the prose was taking a lot longer, but now I realize I can just mass convert it to lean and put it out there.
The fact that ETH is false when many people expected it to be true is the amazing thing. This isn't AI finishing the last 10% of a proof and stealing the credit--this is AI exceeding human thinking. If nothing else, this buries the "stochastic parrot" position.
That said, AI is spiky, and I fully expect there to be a class of problems that AI is really good at solving and a (maybe larger) class where AI is worse than human mathematicians.
I'm incredibly skeptical about everything these frontier labs put out. Results are results though.
However, could someone with a firmer grasp on the problems weigh in on how much are these recent results are function of cutting a virtually unlimited budget for some very smart people? I believe the LLMs certainly help iterate on ideas faster... but I am aware of very impressive work that goes on in quantitative finance. Stuff that would not really be published. That stuff is the result of paying smart people millions to produce results which produce $$$.
I have to wonder, would this have been solved if we simply gathered tons of smart people and paid them a ton of money to solve it? Or is it exclusively because they had access to LLMs?
Could they have achieved this result with previous versions of ChatGPT?
Quant finance is very different from math research; quant is much more applied math and programming, and the tasks are typically smaller than math research questions. Although it's true there's a huge overlap in the people good at both, there is massive re-training needed if you want to turn non-math-phd quants into research mathematicians.
As I argued in another comment here, I think even with unlimited budget, humans wouldn't be able to reproduce these results within a few years. I think AI has crossed a superhuman theshhold in math problem solving skills.
> We do not seek academic credit for the discoveries produced by our internal models.
Sure. But that isn't what they really cared about anyway. They care about notoriety among the general population, and especially among people who decide to pay for their products.
I do not like this at all. At best, we find that there are classes of problems that even AI cannot help much in solving, in the sense that the rationals are an infinitesimally small part of the reals. At worst, a fundamental result trivializes a problem whose complexity is a key component of a critical system, and there’s societal upheaval. If PhDs ran the government, and thought the latter, we might have already seen military attacks on data centers.
Honestly, I’m a bit scared but also a bit excited.
I’m currently reading Africa by John Reader which covers a lot of humanity storyline. From HN I get the humanity achievements that are happening right now.
The feeling digesting both daily is in order of magnitudes bigger than what any really exciting tv show did to me.
I honestly at this point don’t know anything about what could be possible and what couldnt.
I understand that this is really freightening, but also especially ETH being false gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits. I know I’m the minority, but these answers really interest me.
Nobody is solving colonization of planets in this lifetime unless we find a way to accelerate mass faster than the speed of light (and decelerate it at the other end) or extend the lifespan/radiation resistance or can preserve bodies in suspended animation for long periods. Even then, that just removed one of the many technical hurdles. We don't know which planets to target- are you going to send people to solar systems when you don't know if they are habitable? Modify people to be able to inhabit otherwise hostile worlds? How do you deal with boredom on a 1,000+ year trip?
Even with aging, our efforts are better spent increasing quality of life across humanity than helping a few rich people live to 150.
> Nobody is solving colonization of planets in this lifetime
Eh, speak for yourself.
I'm not sure how anyone can say "nobody is doing x.." or "we'll never be able to do y.." after the advances of the last few years. Not sure you understand the implications of what we're seeing here and what RSI/ASI will bring about.
Even just imagining the technological acceleration we'll experience when open source models at the Opus level can be run without insane hardware (I guess in a year or so) or if we just stopped development of AI today - is still insane.
If there's a will, there's a way. There are engineering solutions to every problem you mentioned.
"Sufficiently advanced technology is indistinguishable from magic."
I've worked in ML since 1994. I've been very impressed with the progress of models lately. I read "The Coming Technological Singularity" when it was first written and dedicated my career to seeing these sorts of things coming true.
That doesn't mean that their exponential improvement will change faster than light travel. Therefore, to deliver a human being to the surface of another planet (let's say Proxima Centauri at ~4 ly away) with enough resources to create a stable colony means we'll travel at less than the speed of light. Even with a solar assist, the best we can do right now is less than 1% the speed of light in an unmanned probe. People are working on alternatives, but those are not the sort of technology you just pop in a space ship and send for 400 years to a neighboring star. How do you support a population for 400 years?
As an alternative, let's just make humans like tardigrades, capable of shrivelling up and surviving high-radiation, low-moisture environments. Then we can take 10K years to get to a remote location. As a bonus, the resulting humans can survive in a much wider range of conditions so we have many more target systems to consider. Oh, but that's human engineering- are you sure that we'll just be able to genetically modify people (technologically and morally) at that level any time soon? A single person died in an early gene therapy trial and it shut down the field for 10+ years.
Basically, what you're arguing is that exponential growth in AI (not a given) will unlock huge material sciences and biological advances. While possible, it's unlikely to combine in a way that suggests "we should send people to other solar systems".
How about sending robots to a habitable planet with biological material of humans, programmed to tell us the story of earth?
Re life quality: Sure technical advancement can lead to cheaper electricity, better materials discovered, disease prevention and a lot of other stuff. A lot is hold back by complexity.
I don’t want to make a call that this will happen and that it’s a net-positive for society. I’m just saying the technical advancements is interesting and enablin.
biological material of humans? you'll have to expand your argument here. Are you reconstituting the material on the other end? If you have that technology, do you need a robot to tell a human the story of earth? Why not just upload it directly?
Like we haven’t solved it but it we partly did finish birth in an artificial womb and we have birthed extinct species through a different breed.
Theoretically it might be possible to do embryo arks where you would send machines to a new habitable planet and they would travel for hundreds of of thousand of years then build everything up and then artificially bring humans back to life.
Whilst this obviously has an enormous amount of challenges, we don’t know a physical law that is not allowing that.
Imagine what we do now and what we do in 100.000+ years when we face extinction.
We might also just blow us up, even though I think extinction is very unlikely
> gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits.
There are infinitely many more ways to kill and torture a human than to save one.
Yeah I think depends on the fidelity of the world model.
If you count “impales person with blue stick while distant asteroid is in position X Y Z” then yes, there are probably infinite, so it depends how you class a discrete method
> In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach
Mathematics and computer science are turning into an empirical science
If all you care about is whether the statement is true, then yes.
But what they're getting at is that we also want to gain an insight of "what is really going on", and it will take time to distill this proof into a theory that gives us a more complete understanding.
"In our view, these manuscripts are best understood as a form of raw research data."
They are staying on top of the exact formulations that spread like memes. I think this one was unfortunate. It's an analogy for a new type of thing that stops attempts to figure out what thing that is.
That said, uh, let's go? Let's see if OpenAI earns some goodwill here. It probably can't go worse than the last math drop!
It is raw research data though. Human math is way more refined. LLMs are superhuman at deduction, yet cannot create their own pre-training (where most learning happens) data.
I looked over the announcement for any substantial contribution, and found they're donating $5 million to a research foundation.
$5 million might sustain 5-10 PhD students for 5 years.
Maybe it's just me but I think they should contribute more in exchange for forcing the scientific community into paying for tokens to generate slop and then pouring their time into verifying or falsifying the slop.
I said the same thing in another thread and someone compared it to no one forcing you to own a smart phone. I will preempt that argument by saying the community is motivated to verify these results because they are important to the field.
I'm going to be the cynical one and offer this translation: "Our results are too important to wait for the pesky peer review, so we are dumping a whole ton of money in a fund and giving some elites a breathless first look so our buzz keeps getting jacked up."
The reality that Science is a grind and there are thousands of new results every year discovered by regular humans who use a regular human process to cross-check and review.
Treating every result from the AI like a gift from god is getting tiring.
These are not run of the mill results from AI, those are surely being posted to arxiv constantly by normal researchers without any fanfare.
These are really big results from AI, if a human had come up with them we would also be making a really big deal out of them too (maybe not as big of a deal, part of the excitement is that this was from AI and more results of this magnitude may soon follow).
Regular human processes, it turns out, have been a bottleneck that AI is now removing. AI is now routinely popping problems that have routinely stumped humans for decades or centuries.
Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.
ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.
QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.
Is there any obvious path from these results to any sort of physics, chemistry, or biology breakthrough? From what I can tell: no (but I lack the physics and chemistry expertise to say anything about the QCD finding).
Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?
From the physics, not without more breakthroughs, and even then I can't imagine what that would look like. The particle physics that you can use industrially has long been understood, and the field has long been long stuck because the only things we don't understand are also insanely expensive to experimentally access. Maybe once we move up the Kardashev scale, though.
ETH sounds more like there's a path it could become important? In general, the recent complexity results suggest that a) our intuitions about that world are unreliable, and b) the truth seems to be insanely complicated. I mean, new faster multiplication algorithms in 2026, what the hell? How can we say anything about the full space of all possible algorithms?
I wonder if the best statement we can make for future AI discoveries is that, ideally, they will allow us to move up the Kardashev scale faster.
The new faster multiplication algorithms are both galactic, are they not?
It does not seem surprising to me at all that AI can find new galactic algorithms that are not of any direct use (at best they can guide us to find better solutions that are non-galactic).
What would be surprising and interesting is if AI can find a new floating point multiply-add that is 2X faster per operation than the current fastest chip implementations. That would immediately have billion+ dollar implications. The teams who already work on this have excellent approaches and invest enormous amounts into tiny improvements already; there are probably not much low-hanging fruit there (proponents of using AI for chip design improvements have impressive claims, but unclear that any of those really worked out in reality).
We already do have a very good understanding of QCD, it’s a very successful model with 70 years of experimental results behind it. This proof may offer new insight but most likely will just be confirming what we already knew.
Why haven't these theories had more impact on materials science? We use DFT and other greatly simplified approaches (with tremendous results) but still I suspect we are massively underperforming in terms of producing materials with useful properties.
QCD simulations (like lattice QCD) are extremely expensive (exponential in the number of constituent quarks) and usually things like material properties require bulk physics, in large system sizes. The separation of scale between a theory like QCD and something like a simple metal makes it very difficult to practically do a first principles computation.
Glad to help (I only really understand the implications of ETH well enough to summarize myself without assistance haha). I figured if I wanted simplified context, others might too.
Title is no doubt crafted to get ahead of the usual online backlash, but here’s the actual announcement:
> We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
1. Local smoothing and the Kakeya conjecture
2. The Exponential Time Hypothesis is false
3. Existence of quantum chromodynamics and completions of the Standard Model
AI is becoming incredibly skilled at tackling problems that once required extensive computing and complex calculations. That said, it seems the community has concerns about their passion and purpose, as these tasks used to define careers and take significant time. Now, with machines handling it all, the role of scientists and mathematicians feels uncertain. Even great minds are feeling discouraged, raising the question of whether this is truly a win for mankind—and at what cost.
I asked Claude how will ETH being solved is a major breakthrough for mankind:
“
Honestly, nothing yet. This is a theoretical result, and it doesn't solve a real-world problem today.
*Why it won't change much in practice:* A running time like 2^(n/√log n) is "subexponential" on paper, but for real inputs it's still astronomically slow. For a few hundred variables it's no better than the brute-force methods we already have. Hidden constants in the exponent could also make it worse in practice. Real-world SAT solvers (the CDCL kind) already handle millions of variables on practical problems using heuristics, and this result doesn't replace them.
*Where SAT shows up in everyday life, indirectly:*
- Verifying that chips (like the one in your phone) have no design bugs
- Scheduling and logistics, like timetables, delivery routes, and airline crews
- Software package dependency resolution
- Security analysis and cryptography research
*What it could change over time:*
- *Research direction:* Many problems were assumed to be hard because ETH says so. If ETH is false, researchers may find faster algorithms for them, and some of those could eventually matter in practice.
- *Cryptography:* It doesn't break encryption, since the algorithm is still far too slow. But it could shift how people think about the hardness assumptions behind some security arguments.
- *Theory of math and proof:* The Lean verification is also a notable example of AI-assisted, machine-checked mathematics.
So the "big deal" is mainly for computer scientists: a long-held belief about the limits of computation may be wrong. Any benefit to normal people would be long-term and indirect, and only if it leads to practical algorithms.“
In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."
(Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)
We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies:
[...]
For the physics result, form an external group of collaborators and defer to them on the release of materials.
[...]
The group requests that all communications with any and all members of the group regarding this effort be directed to the following email account: YANGMILLS4D@gmail.com.
>>> We reached out to external experts and asked them to act as shepherds
Sure, if you meet some resistance to your method of undermining a field, the best go-around is to co-opt the top professionals to help pull the ladder up. Astonishing, the arrogance of whoever didn't bother with this before.
I think OpenAI deserve credit for adjusting their approach after the last math drop. I think Terrance Tao shares some credit here for galvanizing the response.
Including the relevant mathematicians and scientists before dumping this onto the public is sure to go a long way to relaxing fears in those communities. I'm not sure how they will deal with accusations of favoritism but at least they are demonstrating a willingness to participate in a process.
I hope the community can express some positivity that will encourage OpenAI and other foundation labs to continue with this collaborative approach.
Well said. Attribution is important, for obvious human reasons and to direct people to valuable context. But our names will fade fast from history, what matters is the journey of human knowledge.
> It is amazing what you can accomplish if you do not care who gets the credit.
--Harry S. Truman
> It is amazing what you can accomplish if you do not care who gets the credit. -- Me, 2026
Jokes aside, this is a thought provoking quote. I do think it's easy to say as a well known figurehead, when the quote and other actions below you may easily be attributed to you, at least partially. But I do agree somewhat, and I think the context around Truman's time (WWII) is a good example of what humanity can accomplish when the stakes at hand supersede the considerations of workers' rights or credit/ownership. Of course this is also a very favorable phrasing for a period of enormous human suffering. I think there is an arms race analogy to be drawn here, though the current situation and stakes are much different.
Funnily enough, there’s a good chance Truman never said those words; the earliest known attribution of the quote to him is from a Time magazine article written in 1988. It’s not in any of his personal writing or speeches.
Yeah, and it’s amazing that we chose to build our society around maximizing the benefits of taking credit regardless.
(From [0])
'A man may do an immense deal of good, if he does not care who gets the credit for it.'
"QI hypothesizes that a Jesuit Priest named Father Strickland deserves primary credit for this family of sayings. This claim is based on the diary entry dated September 21, 1863 written by Sir Mountstuart E. Grant Duff and multiple attributions thereafter. However, similar maxims may have been developed independently."
[0] https://quoteinvestigator.com/2010/12/21/doing-good-selfless
Isn't this ironically completely against the idea of not caring who gets the credit
> If you can't handle me at my worst, you don't deserve me at my best
--Albert Einstein (Director, Los Alamos Laboratory)
Surely that was Oppenheimer.
Attribution is a fascinating topic.
For example, one generally don’t name cells in their body, and cells works generally pretty well to sustain a whole self.
On societal level, obviously no one is aware of who authored the biggest mass of benefit that is granted by the mere fact of living in society. No specific individual have credits for example regarding the language they use daily¹. For the most recent ones, it’s now often possible to attach some name to invention of some cultural/technological artifact, but of course Zeitgeist-wise perspective tend to consider it’s rarely a single punctual individual that should be considered the critical pivot point.
On the other hand, even Marx that was so critical of individual alienation couldn’t refrain to make the intellectual equivalent of pissing territory and attach a name referring to its own ego within all of his works. Ego is a masterpiece of illusion: it will bring the foundational leveraging to build every possible societal hegemony under the sun while convincing the deluded that serving the ego is the indispensable first step of everything to best serve every individual selfs.
¹ Even for constructed languages, such as Esperanto that I practice myself, whoever construct them obviously don’t start from cultural blank slate.
I don't understand. The AI-math guidance counsel has recently explicitly argued against what you say in paragraph two, and which is why for the Kakeya problem there was no group formed. The different communities have given different suggestions. I'm not sure how much I agree with the first paragraph either given these facts.
The approach hasn't changed all that much. It's more that the scientific community is slowly starting to accept the inevitable.
From the article:
> We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies
1. They identified relevant experts for each of the discoveries they wanted to release
2. They collaborated with them to determine how to release the results
The effect for the first two (maths, comp sci) was the same because the experts they chose made that decision.
The downside here is actually the third case (physics). Since they only chose 5 scientists (David Gross, Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov) and those scientists chose to keep the results private until they finish their own work. Everyone else who might be interested has to wait.
That was my favouritism aside. Scientists working on similar or adjacent problems that are not included in these kinds of private releases might feel they are not getting a fair chance.
This is a process that will change, but it seems they listened to the anger from the math community and made changes that help address the concerns.
I think, sadly, these fields will collapse further into a tighter in-group. Anyone outside that group will find it increasingly hard to penetrate. Talent won’t get you there anymore.
From my experience in research this was already the case 15 years ago (and from older peers, way before). Science has always has "in-groups". I forget the name of some philosopher in the 50's who was the opinion science is just a bunch of fads and it takes the old generation to die before a new idea can really be accepted.
“A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” Max Planck 1949
Commonly paraphrased to: The knowledge quest proceeds funeral by funeral.
I'm happy to relax for six months, and ask Qwen-8-E2B on my Nintendo 3DS for the solution.
The hubris in the framing is astonishing, amusing and sad all at the same time. Many say, that's what HN attracts. Anyone who talks about inevitability except of death in these times is just someone who considers they'd be in the "have" rather the "have not" category.
I still can't see what this means for human understanding and how open would this knowledge be. Who'd ultimately get access to this stuff going forward and what kind of power games would ensue hereon.
I'm not sure why you think that someone who believes that this is inevitable is just someone who assumes they'd be in the "have" rather the "have not" category.
It seems like you are doing exactly what you accuse the parent commenter of doing. The only difference, perhaps, is that the commenter seems to extrapolate past progress to future progress, while your own assumption is a lot more tenuous.
What assumption do you think I make? I'm curious.
I just find it uncertain what lies ahead. I just don't see the sustainability in this. I have degrees in Mechanical Engineering, Applied Mathematics and Scientific Computing. If the "inevitable" is that I do manual labour till we can augment the robots to do that as well, I don't see how me or millions others similar to me have any agency.
Right now, a bunch of stuff is blurted out every month or quarter and people in notable positions are given a chance to "tidy it up". Let's assume that you have even the open weight models at your disposal, the hardware to run any of this in the kind of scale needed, requires capital of Governmental level funding.
Where I see this heading is that there'd be a divide of who controls the resources to produce and who gets delegated to do the menial work. If that's what you mean by my tenous assumption, please do share what you see differently.
Your assumption was that someone who believes that this is inevitable is just someone who assumes they'd be in the "have" rather the "have not" category. I agree there may be some correlation, but it is less of a sure thing than thinking AI replacing mathematicians is inevitable, in my opinion.
It’s always weird when people are so quick to use a single positive thing to trivialize the many historically questionable things an entity has done up till this point. It certainly doesn’t come from a place of rationality.
What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
> What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
This is like going to a politician's event and seeing him giving away lollipops to toddlers and kissing babies. But it's the same politician caught in corruption and embezzlement scandals.
You look at that and praise how they have redeemed. Other people look at that and see a corrupt politician handing out candy.
People tell us who they are through their actions. If you prove yourself to be capable of morally reprehensible things to the point throughout the years you build a reputation for all the morally reprehensible things you do, does your potential to do morally reprehensible things change if you go way out of your way to let people know that sometimes something's you do aren't morally reprehensible?
Wrong, these companies have shown time and time again they are irredeemably misaligned with our interests.
Right now you and everyone else is doing the "leave the multi billion dollar company alone" version of the neckbeard "leave the lady be" meme. And you deserved to be chastised for it.
Its the equivalent of you going around telling everyone in the office about how your boss is turning it around because they bought donuts today, after they spent three years telling you you're worthless, they're entitled to taking all the credit for your work and they plan on firing you as soon as they can.
Anyone can solve any problem they want and release it. Nobody is defending multibillion companies, they are pointing out hypocrisy when the same would be celebrated of it was some academic mathematician who also used ai released a solution vs open ai or any other entity doing so.
The history isn't particularly long.
This is going to be a very big deal in some of my circles, which include KITP (the Kavli Institute of Theoretical Physics), where David Gross was the Director for many years and now holds the Chancellor's Chair Professor in Theoretical Physics.
I’m not sure what meaningfully changed except for the QCD / yang mills result (where leaders in the field are going to do a review and write up first).
After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
Allegedly.
> After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
This.
The truth if the matter is that this is still a publicity stunt. This investment of time, effort, and compute is aimed at building a reputation that this company's model is better than the competition's. The previous publicity stunt brew up in their face, and now they seem to be trying to adapt and double down.
I would have found it a lot more impressive if they just empowered scientists through some kind of grant rather than to internalize all this. It sets the tone for what's to come and inevitably if what they find is juicy enough it will not be published. For instance once they start working on materials science (if they aren't already).
Surely they’re under some kind of NDA with Defence with regards to material science and physics, and required to hobble the publicly available models.
Why would you think that? Is a privately held company.
Attribution is very important. Though AI hasn't really done anything yet so it hasn't affected actual mathematicians yet.
Yeah it’s just one Millenium problem, no big deal. I was gonna crack quasi-Reimann next year anyway, just hadn’t gotten around to it.
All AI is good for is remixing frontier mathematics, there’s no way it could ever make its own. Right?
I hope it gets us somewhere new. You can hope that and appreciate it hasn't yet though. (I'm not part of their IPO)
It's parasitic on humans and it's not an infinitely deep well. Wrong?
Ok my original comment was snarky (apologies to all three mods), but you've raised one of my favorite topics so I can't resist engaging a bit more seriously! In the second clause, that is -- the first is trivially true in some senses and trivially false in others, so it doesn't seem very productive to nail down.
So Chomsky thought of himself as a philosopher of science more than anything, and his life mission is/was to answer a fascinating question: what kind of creatures are we? There's been a growing consensus going back to at least Darwin that we're separated from the rest of Animalia by our "reason" (!= intuition != understanding != judgement != volition), but he helped popularize an old theory[1] that puts it kind of beautifully: human language is only possible in beings capable of producing an infinite set of meaningful outputs in response to a finite input.[2] And so far, that list includes only two definite members: Homo sapiens sapiens, and computer programs built upon large language models.[3]
So all that to say: yes, I think you're wrong; the programmatic ensembles that brought us today's breakthroughs are an "infinitely deep well". As long as we have questions to ask, they are not theoretically constrained from answering them.
Of course, this invites some obvious follow-ups on practicality; even the most impassioned ASI boomer wouldn't argue that the exact ensembles used here could answer every question in the universe. But A) that's possible because some infinities are bigger than others, and B) the exact same caveat applies to humans, whose cognitive capabilities are thoroughly determined by accidents of biology, culture, and episteme. The latter is the thrust of Chomsky's famous debate with Foucault, which is absolutely worth a watch for anyone who hasn't seen it (starting around ~18:00, AFAIR): https://youtu.be/3wfNl2L0Gf8?t=1080
---
TL;DR: I ultimately disagree on both points, yes. In the ways that matter, machines are no more constrained than animals.
[1]: https://en.wikipedia.org/wiki/Wilhelm_von_Humboldt
[2]: Many animals are capable of communication, often quite complex and fascinating communication at that; even flora and fungi communicate! But language is qualitatively distinct. You may have heard this in the pithy-yet-mindbending sentiment "we've taught many apes to communicate with sign language, but none of them have ever asked a question."
[3]: It's no surprise in hindsight that trying to build better autocorrect got us to AGI; I find it unspeakably tragic that Chomsky never got the chance to see how thoroughly he's been vindicated, in this regard. He had his stroke mere months into the singularity, and had a quite negative view of the approach before then.
> human language is only possible in beings capable of producing an infinite set of meaningful outputs in response to a finite input. > And so far, that list includes only two definite members: Homo sapiens sapiens, and computer programs built upon large language models.
In LLMs that's a bounded infinity though, the models themselves have only so many bits and the activations are also limited. Once you reach the same state twice the model is looping and it won't break out of that loop. I've seen such loops across tens of thousands of tokens so I'm not sure to what degree the infinite output rule holds true. It would be an interesting experiment to try to figure out to what extent the 'busy beaver' can be kept busy before it degenerates to looping or nonsense.
Humans are finite bodies.
LLMs are just regurgitating an output according to a set of rules generated from harvesting all human output.
Language to humans is a symbolic mapping of labels to reality based on actual experience.
"Language" to LLMs is just a string of tokens to manipulate based on relationships harvested from recorded human experience.
It doesn't mean anything other than 1s and 0s - it's not a language because the LLM isn't communicating with you, just as the sum of humanity isn't communicating with you.
It's just a really good "choose your own adventure" book.
...do you think molecules and their interactions are also not bounded physical systems? What part of a human being do you think isn't a turing computer? Please release your groundbreaking physics paper proving this.
Apollo the African grey parrot not infrequently asks “what this?”
He may just be parroting something he heard.
They don't necessarily deserve credit, it is just the bare minimum courteous behaviour in academia to reach out to other researchers in the same area before publishing a result.
I'm also unaware of the quality of the consultation with the relevant mathematicians, colour me skeptical there.
Here let me fix their statement for you:
> "The external collaborations will follow four principles... until Anthropic, Google, DeepSeek, Google and dozens of Chinese companies all make the same offering, in which case we will rediscover and re-implement all of the things you hate about traditional academic advancement and publishing, except because our companies are run by people even more mercurial and self-absorbed than university presidents, we'll wind up supporting even fewer people to do it."
ETH is false! I really would not have believed it.
[0]:
> We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula. For every fixed k ⩾ 2, on a formula with n variables and m clauses, our algorithm uses $poly_k (n, m)$ space and runs in time $(n + m)^{O_k(1)} 2^{n/(\log n)^{1/2 + o(1)}}$
The dominant term there is (roughly) $2^{n / (\log n)^{0.5}}$, so still superpolynomial but less than exponential. Fucking wild.
[0] https://github.com/openai/math/blob/main/preprints/The-Expon...
And the claimed algorithm solves the counting problem, i.e. determining the number of different solutions, which in many cases is much harder than just determining whether a solution exists or finding one of the solutions.
They gave the algorithm to the original authors of the papers to review, I bet their reaction was similar to yours.
Yeah, this really blew me away. I took a graduate computational complexity course and the original prof had to drop out due to health reasons. We got another professor from a completely different field who was basically as green in the field as we were, but with far better research chops. It turned the course into kind of a long-running lab where we just kind of did whatever we wanted. We spent a lot of time just philosophizing (mostly uselessly) about the ETH. I am very chuffed to see this result.
And who says AI doesn't create open problems. This unlocks thousands of high quality open problems.
Turns out the 'opne' in open AI wasn't referring to open source, but open problems!
It's not AI that has created those open problems. It has simply shown that some conjecture is false and that leads naturally to problems in a sense already posed.
A very different thing is proposing a framework and in it identifying interesting questions. So far AI has only worked within existing frameworks addressing problems already posed. That may change in the future, but it would be a genuine and qualitative evolution.
ETH is false! This is one of the big long-standing hypotheses that I actually remember from university, and afaik, it was reasonable to assume it would be true; reasonable enough for a lot of papers and results to assume this for bounds or hardness proofs of common problems in parameterized complexity, anyways.
Also glad to see that one of the authors of the paper stating it is also in the loop here.
2026 is weird.
What were the reasons to assume it would be true? I mean if you assume that NP < EXPTIME (which seems very likely) then you end up with algorithms that run in exponential time, but can't simulate turing machines with polynomial overhead. Obviously you can't simulate TMs with just anything, but most of the time... computation finds a way?
What will happen to fine grained complexity if ETH is false? So there will be a solution for the Clique problem? So our social networks will evolve differently after this?
From what I understand, this proves mathematically that complex problems can be solved faster than we thought.
Now... the algorithm only works at impossible scales, so it changes (probably) absolutely nothing for real life or social networks.
Impossible scales in 2026 terms but what about 2076? Genuine question
Interesting that the physics result is not released yet. From the limited wordings, “unified completion of the Standard Model of particle physics” is probably SU(5), which is experimentally disproved as one describing our universe. So the wording sounds a bit confusing, ie they are not proving or discovering Grand Unified Theory (GUT), and there’s no Higgs. So it is not really the standard model, but models that’s closely related to that. It is still an impressive result. IMO at the level of the recent quasi Riemann hypothesis level result, shy of getting the millennium prize (no mass gap) but is actually a very meaningful result, both of which arguably more impactful that the one they solved a millennium prize problem (Navier–Stokes).
More comments in https://blog.kolen.dev/link/openai-shepherds
100 years of so close yet so far, although satisfying, would be humbling if a GUT was finally discovered not by humans.
Capital allocation to advance mathematics and sciences is undoubtedly a good thing. But this does beget a few questions:
1. How much did it cost?
2. Would an equivalent allocation to incentivize human effort have borne the same results?
3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
#2 is particularly important to answer, given research grants have been getting cut elsewhere: see U.C.L.A's research grant cuts from last year and Terrence Tao's attempt to raise funds (about $60k) by collaborating with Open AI earlier this year [1].
[1] https://www.nytimes.com/2026/10/16/us/math-ai-terence-tao.ht...
As for #2, personally I feel like OpenAI's results are superhuman at this point. I don't think there's enough human talent alive right now to produce these sorts of results within a few years, no matter the budget.
There are lots of reasons I believe this. One is that frontier AIs are very clearly superhuman on shorter problems (as can be seen on benchmarks like FrontierMath, or just by playing with them yourself if you have a source for difficult problems)--so these achievements are "just" scaling that advantage to research-sized problems. Second, AI is also clearly superhuman in its knowledge across fields, so it can make connections and combine techniques that humans won't be able to access. Third, these problems are famous and have withstood efforts of many of the best humans, in many cases being considered unapproachable.
Altogether it seems like AI is not only smarter but also more cost efficient that humans for math-adjacemt problem solving. I'm not saying that AI is superior at everything in math, eg theory building, but I think it's really unlikely you could get any group of humans that could compete with OpenAI's latest results.
It depends on the definition of superhuman. For example my car is superhuman as it allows me to get in one hour to a place where it would take me ten hours to go by walking, and can I walk ten hours non stop? However it is not superhuman enough to carry me where I could not walk too. And there are places where I can walk to but I cannot reach by car. Maybe that applies to AIs as well. A plane is in the same category of superhumanity of cars, with more speed. But I could still get to Australia by walking and rowing with my hands on a log (and probably die, I know.)
However there are truly superhuman vehicles, spaceships and submarines, that can carry humans where no human could go alive.
I don't know if these AIs are superhuman in the way of planes or in the way of spaceships. I also don't know if they can do everything we can do by talking or if they have some limits.
I mean, a plane can fly but not reach a branch and stay still on it. A small drone, almost. My coding agents sometimes can't do the obvious but is it because of some intrinsic limitation or because my instructions are bad? So far my experience is that they are superhuman like cars or trains, not yet as planes.
>1. How much did it cost?
>2. Would an equivalent allocation to incentivize human effort have borne the same results?
Unlike their initial massive brute-force run that solved the Navier–Stokes Millennium Prize Problem, most of their subsequent breakthrough results in mathematics have come from a single prompt to a single AI agent, using approximately 3 hours of ChatGPT Pro-equivalent thinking compute per result on average.
Find me a Grand Unified Theory, make no mistakes
1. some bad hack for the switch2 took 2 weeks of non stop work for an AI, so most certainly this was expensive.
2. most probably no, the AI doesnt rest and its very very fast.
3. also no, they are doing this for aura farming which then will result in more money for them, the money will dry as soon as this stops working.
The problem with point 2, is that person couldn't then work on other things. And there's a limited supply of people of that capability in the world.
We're now at stage where we can turn energy into thought to generate expertise on demand.
I think you are interpreting 2 as offering some large amount of money as a prize for discovering a solution. But I think it was meant more as funding a team of mathematicians to work on the problem.
> 3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
I think the scarier alternative may be OpenAI forever gating their super-frontier models, capturing a significant portion of the sciences and leveraging it for equity in businesses that build on scientific advances. e.g. "Hey CATL, how about you guve us 5% equity in exchange for a battery chemistry breakthrough our AI has found?".
It's roughly equivalent to the playbook Palantir had for the public sector when they marketed themselves as a company that had cracked data science. At least that's what they've been pitching back then. I can't find a lot on public disclosed deals based on that.
1. it comes from VCs philanthropy, who cares?
2. it wouldn't happen so who cares?
3. no, but models will be better, it'll be cheaper to do
It's not a snark. The scale of VC philanthropy is unprecedented and it fuels itself. All of it will eventually, one way or another, become open. Long story short – there is nothing to complain about.
or it will fail spectacularly (also not sure philanthropy is the correct term)
success and failure of anthropic are both good for open weight models
Just wanted to say that these questions would indeed be great to have answered, or at least discussed. It's weird how cultish some people get.
In 2025, Hong Wang won the Fields Medal for proving the Kakeya conjecture in three dimensions. Here, OpenAI is claiming to have solved a stronger problem that proves the Kakeya conjecture in all dimensions!
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
To be honest this doesn't feel as such impressive as the others. It was recently proven in 3 dimensions, so it could be "just" an extension of that. I haven't read the paper anyways so this is pure spectulation.
The question for me is, would OpenAI have been able to solve this without Hong Wang's work?
The question for some is, is whether the original authors would have produced their work without building off from the work of other mathematicians.
(which is somewhat valid). Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
:/
IMO that there is any work at all to build from is the result of a long tradition of taking the next step. Professor to student and so on. The danger of AI here is we get to take a leap, say, 10 steps - but then there is no more tradition to have the fortitude and mentoring necessary to continue the progress.
It all gets derailed while betting on RSI and never having to do math research again (but if an advanced theorem is proven, and no one alive is competent enough to comprehend it, does it even matter that we get to call it solved?)
This is why the AI companies are making money off selling tokens and raking in investments for as long as it will work and is not looking to create the value themselves and making money off it...
They know that value creation is a dead end sooner or later..
If we suddenly skipped 10 generations of mathematical lineage because of AI, wouldn’t it also be smart enough that you could just prompt it to connect ideas from graduate level maths all the way to the proof!
I think the question is rather, what will happen to human beings when any possible personal intellectual accomplishment will no longer be seen as interesting or relevant by other human beings when AI can do most of it?
It's all well and good to say that we should just go on these journeys for our own enrichment, but even so they will just be doing it alone...or else not do it at all because they need a shared human effort surrounding them to do it, and therefore end up depressed and immersed in drugs or media all day.
Of course we're awed at what AI can do now, because all of us observing it have had the luxury of having such opportunities before AI. I got my PhD before AI and I found satisfaction in participating in a shared math community, which I wouldn't have done if I were starting my PhD now, because I would find the current academic environment unacceptable and useless.
Many people still do woodworking for the love of the craft, even though factories exist with machines that can do everything hand tools can but 10x better and faster. If AI gets way out in front of anything humans understand, I expect that mathematicians will probably pick up where their understanding actually is, and carry on.
That analogy is rather poor. Woodworking produces discrete, unique, and physical objects, that can be made, and found use for, as many times as the craftsman wishes. Once you discover a law of physics, you can't discover it again, and it's not yours to do with as you please.
> Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
I don't think so, it was a big leap forward in terms of creativity, imagination, rigor and experimentation. Einstein was able to "run" visual thought experiments inside his mind
That would be a very exciting experiment. I doubt you could find enough vintage training data to train a LLM enough to do it without it being contaminated by anachronistic data in the weights, but I would be absolutely amazed if it could reproduce that.
whether the original authors would have produced their work without building off from the work of other mathematicians.
That's rhetorical, right? Of course mathematicians built on what older mathematicians built and OpenAI's machine built on that. The real question whether OpenAI's machine has contributed something others will in turn build on.
Not saying I agree with the view, but something along those lines popped up when the Naive Stokes conjecture was seemingly solved (and way before when Open AI began solving Olympiad level problems and so on with their older class of models). Personally, I do see both sides of the argument. Though the broader impact remains to be seen.
The proof uses completely different methods as what harmonic analysts used for the Kakeya conjecture prior so probably yeah.
Would Hong Wang would have been able to do his work without previous physicists work? That's the real question!
*her work
Yes. The all-dimension result follows a completely different approach from that of Wang and Zahl.
I mean, the models (starting with Bel clearly and Astra probably--though Astra probably is worse than top humans in some areas) are superhuman at math.
I understand that the Exponential Time Hypothesis being false doesn't necessarily mean P=NP, but for people who are a lot more knowledgeable about complexity theory, does this update your priors significantly in the direction of P=NP?
It's been a long time since I seriously thought about complexity theory, but I'd say no. For similar reasons to why complexity class E is not closed under polynomial-time reductions.
The most important component in the time bound is of the form 2^{n / polylog(n)}, which is subexponential. But if you have reduction from another problem that increases instance size quadratically, the algorithm for the other problem is no longer subexponential. Which means we are still essentially in the same neighborhood as before.
It's a huge result if true, and probably the first major AI result that affects the papers I read. But since the boundary in the hypothesis was kind of arbitrary, breaching it should not significantly change our intuition on complexity classes.
I think the bigger issue (for someone hoping for P=NP) is that there's a huge gap between "polynomial" and "exponential." Even if it were the case that all NP problem classes had subexponential decision algorithms, that would not get us close to P=NP.
How much effort has been expended on optimizing those reductions though? I have a harder time believing that one can hang one's belief in P!=NP on that particular hook.
The challenge is the initial reduction from an arbitrary problem in NP to the problem you want to prove NP-complete.
In the usual approach, you assume a nondeterministic Turing machine (or equivalent) that solves the problem and a time bound (which can be an arbitrarily large polynomial). You then create a boolean expression that can be satisfied if and only if the Turing machine accepts the original problem instance. Unfortunately all constructions so far have included the time bound as a factor of the size of the expression.
There are so many just barely sub-exponential function shapes between 2^(n / sqrt ln n) {this result} and 2^(log n) {polynomial time}!
I think that it'd be funny if the polynomial time SAT solver was just "send your SAT instance to Astra and have it try to solve it for a few hours."
I wonder if there are proofs regarding what neural networks like the transformer can solve. Because fundamentally, Astra is just a bunch of matrix multiplications and a few well defined non-linear functions.
Anyways, most of the times, SAT can be solved quickly, but not always, so getting the answer from Astra in polynomial time doesn't prove much unless you can also prove that it always does, no matter the input.
Iirc, there's a big difference between the computational complexity of what a transformer can solve and what it can solve in practice; what it can solve in practice tends to come from things like singular learning theory (which describes the forms of generalizations that a given ML algorithm can make), but I think (iirc) that transformers can compute a pretty broad class of O(N^2) functions?
For the umpteenth time a transformer is an arbitrary turning machine due to the fact that it has a theoretically limitless tape (the context).
A single transformer step is simply O(1) (for a particular parameter size), but it applied a potentially infinite number of times.
Thus while a single transformer cannot solve any problem, a transformer system applied in the normal way can solve any computable problem.
The context acting as a "theoretically limitless tape" while simultaneously the step being O(1) are contradictory. You cannot have it both.
The context can change each time. To implement a UTM only requires 2 states and 3 symbols or the reverse.
So if you use the context not as a tape in itself, but as an io channel to read/write an infinite tape, the context for any given step can remain fixed in size, and so the step itself can be O(1).
It's not contradictory at all. The dfa at each step of the turing machine model is also o(1).
Of course it is. You’re conflating a turing machine with a transformer. Transformers don’t have a local read/write head. The [self] attention is over the whole context, which makes each step O(N) at best (with caching) and O(N^2) w/o.
Imagine you're trying to prove that water is wet (P!=NP). You lack a formalism that could make sense at building a proof even though pretty much everyone agrees from empirical evidence that the water is indeed wet. A promising attempt at building one such formalism was just demonstrated by a model to be a dead end.
I like this, but the actual finding is a bit different. Trying to keep the water is wet makes this awkward, but I believe it's more like:
Everyone agrees that water is wet (P!=NP), but no one can prove it. In the meantime, a whole field of computer science was built on a stronger belief (ETH): that water, at its worst, soaks. Turns out, even at its worst, water only makes things damp. It's still wet. That said, this finding did produce a new measurement of wetness.
At certain temperatures water is indeed not wet.
A nontrivial % believe P = NP with some reasonable probability. I wouldn’t say it’s all that settled unlike say RH
Anyone credible? A nontrivial number of people also believe that humans used to ride dinosaurs around like in the Flintstones.
> Anyone credible [who believes P=NP with some reasonable probability?
Do you accept
- Moshe Y. Vardi [1]
- Anil Nerode [2]
- Donald Knuth [3]
to be credible? Source for the statements that they consider P=NP to be possible: [4]
And you may include me in the list of credible P=NP believers because treat myself as a credible authority, i.e. when I ask myself whether something is right, I consider myself to be much more credible than some famous scientist. ;-)
(to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems. One should consider the evidence by oneself instead of believing in P!=NP because lots of computer scientists claim all the time that this is likely true, or consider P=NP now to be possible just because some famous computer scientists consider it to be possible)
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[1] https://en.wikipedia.org/w/index.php?title=Moshe_Vardi&oldid...
[2] https://en.wikipedia.org/w/index.php?title=Anil_Nerode&oldid...
[3] https://en.wikipedia.org/w/index.php?title=Donald_Knuth&oldi...
[4] https://en.wikipedia.org/w/index.php?title=P_versus_NP_probl...
Note that, while Knuth believes that P = NP, he doesn't believe that it will have practical applications because the exponents and/or constants will be galactic.
> to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems.
No, it's an irrational rejection of the notion of expertise. The experts are not "heroes", they are simply people who have done vastly more study than folks like you or me. e.g., if Magnus Carlsen looks at a complex chess position and says that white is winning, I will trust his judgment far more than my own.
> The main argument in favor of P ≠ NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration. — Moshe Y. Vardi
That sounds particularly relevant in how AI-assisted/driven mathematical research is changing the game, where previous assumptions based on human experience/effort are being overturned unexpectedly.
stack overflow worthy levels of ego
Not the best analogy; water is famously not wet; it makes a surface wet is the common consensus ; wetness is being a property of a surface
No, it is famously a pointless internet debate with no settled answer. There is no scientific definition of what can or can't be wet, saying that wetness only applies to surfaces is just an opinion. It's an entirely linguistic question, and most people would say water is wet. If you want more than descriptivism, the common definition is something like 'containing or being covered with liquid', and I think most people would agree that a mass of water contains water.
I'm beginning to wonder whether P!=NP is undecidable.
Water is not wet. Human lacks water sensor. All the sensors we have a temperature ones so we sense "wet" as a combination of temperature and visual information. Close your eyes, touch water of your skin temperature and you won't sense it. It's not wet.
Stick your head in it and breathe. You'll know.
Not only is that statement not true, as noted by others, it's also not at all relevant to what you're responding to.
Sand too, even when dry. But OK, metaphors have limits.
By that definition, a vacuum is wet.
There exist lots of non-wetting liquids in which humans cannot breathe.
Fog, clouds, steam, ice.
It might make the third option, that P vs NP cannot be proven true or false, somewhat less likely.
I think it would have to if you're honest with yourself. Any proof of a novel implication of P=NP, such as !ETH, should increase your belief in the two sets' equality. Just as any novel implication of P≠NP should adjust your belief the other way.
My thesis was around showing implications of few different hardness conjectures. So grounded was ETH that I introduced it in a side note as Conjecture 0 after Conjectures 1 and 2. I don't think it's easy to express how substantial of a Bayesian update this is all around.
No. Scott Aaronson said that it changed it from 3% to 5% for him, which is a lot more generous than I would be (but he knows far more about the subject than I do).
So, what I heard was "No, and here is a good reason it should be 'yes, some'."
The question was
> does this update your priors significantly in the direction of P=NP?
My answer is no ... and Aaronson's answer would also be no ... clearly he doesn't view a change from 3% to 5% to be significant ... his response downplayed the significance of !ETH in re NP?=P. So apparently you downvoted me for simply stating my own opinion while you completely misunderstood what it was an opinion of.
A change from 3% to 5% is significant. It's almost doubling the probability.
For me, this result is significant. It basically adds much more hope P is equal NP (that majority seems being against). I believe one day AI will explain a major flaw in modern mathematics, so a kind of correction in axioms will be suggested to prove Rieman's hypothesis (or a completely new apparatus)
I don't know if that's a "flaw". It's what Godel Incompleteness guarantees.
Would be funny if the most famous problem turns out be an example though
That's not that how that works btw. The incompleteness is a property of the algorithm which does the proofs, not the theorems. If a theorem is true it has a proof and a program which proves it (printf("the proof verbatim")). But no computer program can successfully prove all theorems. The space of potential proofs is too big to search systematically, but you can still get lucky.
That is also not how it works. The theorem says that there exist some true statements for which there is no proof. This is assuming the axioms are consistent.
I think you're right, I'm conflating two separate things.
EDIT: I am wrong (it's been a while) but there is nuance.
I said any computer program fails to prove some true statements. In fact, some 'true' statements cannot be proven by any program, as others have said.
But for those statements G, you can prove 'G is true if our system is consistent', just not 'G is true'.
So when we say G is 'true', we are really just asserting the consistency axiom that our computer program is denied.
Godel says that there are statements that are true but have no proof within that system. It's not about algorithms.
I'm saying it'd be funny if Riemann were true but not proveable.
Though I guess it's actually funnier. If Riemann were false then there would be a counterexample, so it's clearly within existing axioms.
So if one can prove that Riemann isn't proveable in our current axioms then it has to be true.
Can something be both true and unprovable without being a self-referential Godel sentence? I assume most statements that aren't self-referential are independent, like the continuum hypothesis with ZFC.
Godel sentences are only true if you assume consistency. And once you add that axiom, the computer can prove them too.
G is independent of your original system T (cannot be proven true or false).
If you create T+ with added consistency axiom, then G is true and provable.
But G is never true and unprovable in the same system.
It's a lot like is/ought or theology.
Why do you "hope" P=NP? It would be a catastrophe
Either P=NP or not. Hoping one way or the other changes nothing. Nor does it change how efficient computing is, nor does it have any effect on cryptography. Discovering a vastly more efficient means of factoring primes--perhaps via quantum computing--would do that. But just learning that P=NP does nothing to further such a discovery. Almost certainly, if P=NP, the exponents and/or constants in a polynomial factoring algorithm would be galactic. (FWIW, Knuth believes that P=NP and that it has no practical consequences.)
Not really. P=NP but the degree of the polynomial is Grahams number doesn't let people into your bank account
it could make lots of compute way more efficient, not sure why it is catastrophe..
Why? Nature doesn't care if people are making financial transactions based on flawed cryptography.
Have you been in nature? It's cold and full of bugs and disease.
> Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026, which led us to develop the collaboration described here
absolutely insane
Why? Did any physicist really doubt it? 4d QCD has been used for approximate calculations for about 45 years
Yes, people did. Well at least I did. We knew we can use QCD to compute loops and to some degree masses in the lattice. But there were serious doubts the underlying mathematical theory was sound. For many it didn't even matter. It would just mean that at those energies the theory is not even valid.
I am not certain on the extend of this proof but QED is probably not sound on it's own and we have been using that even longer. lambda phi^4 behind symmetry breaking and the higgs is strongly believed to be unsound on its own.
Yes, but it has also been known for 50+ years that QED has a Landau pole and might be UV trivial or inconsistent, whereas QCD has asymptotic freedom and is thus controlled at UV.
More to the point, QCD has been experimentally tested both at UV (deep inelastic scattering) and at IR, while the physical meaning of testing QED at UV Landau pole regime is unclear.
they should point their AI swarm at quantum gravity now
(only half tongue-in-cheek)
The problem here is experimental evidence, which is the general reason they focus on mathematics.
I find it a bit ridiculous that they claim three discoveries in "math, CS, physics" when they are basically all pure math (in particular, they all may be formalized in Lean) and only belong to these different disciplines for historical reasons.
Isn't whole CS basically pure math?
So again farming the low hanging fruits for marketing?
Call these problems "low hanging fruits" may be the understatement of the century.
If everything of this can be represented in Lean, with enough compute, they are low hanging fruits, yes
Think openai should start including visual and interactive explainers in these where the viewer can toggle level to get a more accessible/technical explanation as needed. Would also be a great demo, and perhaps increasingly essential for us to know that explainability is not lagging behind provability.
I think at this point openAI does not really understand the results themself and any easy to understand visualisation would imply understanding. I would rather wait for the work group to produce some explainers or at least approve visualisation
Lots of progress recently in formally verifiable domains. Really curious if this is a product of the machine learning folks behind most large language models being much more likely to have a background in math, physics, etc. Or if it is that large language models have some fundamental limitations in the more "scruffy" sciences like biology.
My guess is that in formally verifiable domains you can use RL-esque processes to get lots of good high-quality training data with reward signals (does it verify?), but the same does not apply to "scruffier" work. I would also imagine that the data volume of high-quality scientific work in "scruffy" domains is too low for the large language models to have a good manifold for "high-quality biology research". Would love to be proven wrong if anyone knows of anyone doing great scientific work with large language models in non-verifiable domains though.
It's not just RL.
Many of the harnesses used to search for solutions rely on quick/cheap correctness checks to discard incorrect hypotheses. With something like biology, you might need to run a wet lab experiment to reject an idea. This can add days per iteration, slowing progress down by a factor of 100 or more.
The sad reality is that many computational approximations for biological phenomenon are relatively weak and lack the predictive power we need.
Is there a path where compute gets poured into developing, and then eventually using, computational models that supersede the shortcoming of the existing computational approximations you mention, which thus makes computational analysis and developments possible at rapid scale?
In an area I am more familiar with though still not my domain: when the stiffness matrix and computational linear algebra methods collide, it unlocks a world of computational structural design and analysis by commoditizing the underlying mathematics. This is a simple example, but you get the idea.
Obviously with CFD, mechanics, etc there is always massive work being done to try to unlock new computational methods, faster/parallelizable computational methods, differentiable ones, etc etc, but the general feeling is that the core physics are well modeled (NS millennium problem not withstanding) and are not lacking in predictive power for useful applications. I would have assumed that the same would be generally true for computational biology etc, so just curious to hear you elaborate more. It seems plausible to me that turning the brute force of “the models” (as in, swarms of AI compute) to work on/develop/solve/address whatever representational problems there are in the computational approximations/numerical methods you mention, particularly if that is a bottleneck to performing more extensive methodological/experimental exploration in silico to enable better exploitation/use of limited resources in wet lab.
Again, I know nothing about computational biology, just stating my intuition and curious what your take is. Probably the limits of my knowledge can be demonstrated by the fact that the only modeling methods that come to mind as being vaguely relevant would be, like,… molecular dynamics, maybe discrete element method in a weird way but doubtful, and various pop science understandings of protein folding etc.
Computational biochemistry is what I'm currently pursuing, but I'm pretty new to the field so take everything I say with a grain of salt: Biochemistry is a really really really tricky problem, for a couple reasons:
1. Biology has a very nasty habit of not cleanly abstracting. It's not like we can treat every protein as a "diffuse mass" without losing a ton of important interactions. This is unlike something like CFD where you can treat a mass of water molecules with certain constants. Think of something like cancer: a small mutation to a DNA strand can cascade into a tumor. You have to deal with the mutation happening at an atomic level while scaling to a whole organism (yes I know there's models that can approximate this, but they can't capture all the interactions).
2. This brings us to modelling at the atomic level. To model a system of atoms with the Schrödinger equation, it naively scales O(6^n) per particle. Note that means every nucleus and electron, so if you have say 10 carbon atoms and 22 hydrogen you're already at 10^127 operations. We can cut corners a bit and end up with something like CCSD which scales O(n^7), or cut more corners and scale O(n^3) for DFT. DFT is pretty much the last method where bond forming and breaking is modelled, so after more corner cutting you lose losing reactions, which are important for understanding why something is happening. Most protein folding simulations (moreso prior to alphafold) run a very crude approximation that models atoms as balls on springs (things like AMBER and CHARMM), which obviously can't handle reactions.
3. If this wasn't bad enough, you also can only move through time at about 1 femtosecond (10^-15), so it takes a ton of simulation steps to get even a nanosecond in. Forget millisecond or second lengths where things start to happen.
So, while the math of the physics exists, computationally it's practically intractible. There's some exciting work with something called Machine Learning Interatomic Potentials (MLIP) which approximate DFT using a neural network which are about 1000x faster than DFT, and iirc scale much more linearly (look at things like MACE and orbmol if you're curious). So we're making progress, but we've gone from so ridiculously impossible to pretty much impossible for large systems.
I have a couple ideas about developing custom microchips to make this faster, which is why I'm pursuing this, but there's still a massive way to go!
To add to this I think there are “reductionist sciences” and “systems sciences”. Reductionist sciences are ones, like physics where there are lots of copies of a thing with near identical properties at different scales (the periodic table would be considered an incredibly short list and that seems to be one of the longest list of “different things” in physics).
Systems sciences are ones where each scale has lots of things with unique properties. If you think of “biological things between 2-4m” the amount of unique objects with unique properties and unique interactions is absolutely mind boggling. These sciences generally do not lend themselves to clean highly parsimonious models. The best you can do is coarse-grain a thing as a system (some very roughly defined set of interactions between different things) with certain behavior while accepting that you are losing enormous amounts of predictive power.
I suspect AI will remain worse at systems sciences for a long time because the “possible space” of models for a particular system is much larger than in the reductionist sciences and navigating it takes a human taste and instinct that LLMs don’t seem to have (yet).
One other thought, I think another aspect of biological sciences is just how much effort it takes to do research. Hypotheses are a dime a dozen, but actually verifying these is quite difficult. It takes a lot of patience, time, attention to details, and even then it often fails to replicate.
I don't think LLMs will just magically be able to connect all the dots, because "garbage in, garbage out" still applies. And there's just too many invisible intermediate states, there is literally just not enough information to fully constrain a model. Hence my interest in scaling computational chemistry to get these intermediates :)
Thank you for such a detailed explanation! I was wondering why we don't see MD everywhere and things like protein folding are all based on ML.
Is there anything like a getting-started for working with MACE or orbmol, a 'hello world'? Like maybe simulating Allolactose binding to lac1 ?
Heh. Unfortunately computer science is really the weird one out where open source is the norm. A lot of computational chemistry knowledge is mainly transferred in research labs and through college classes, and you have to pay for a lot of the tools. But hopefully I can at least point you in some useful directions. Here's a bit of an overview of the CC workflow and free options (note I haven't tried all the free options):
1. The first thing you need to do is start with some atomic structure. You can often find pre-existing pieces (through websites like pubchem), but you need to cobble them together. I use chemcraft and gaussview for this, but those are paid options. I've heard good things about the FOSS program Avogadro though.
2. Once you've built the structure of interest, you want to calculate properties of it. Note that we only infrequently simulate the atom's movement, since that's quite expensive. There's a lot of techniques that let us get the most bang for buck, like calculating transition states, or states of lowest energy which we call intermediates, or running something called an IRC, which follows the molecule to its lowest energy point (note we pretty much always think about these things in terms of energy). The traditional programs I use for this are Gaussian and ORCA (ORCA is free but you do have to create an account on their forum). Newer methods include MACE and orbmol, which are pretty much only used for trajectories (e.g. running through time) as far as I know. This newer crop of programs are much more FOSS, but they aren't as rigorous so my lab uses them only as a "rough draft". I pulled in one of my current notebooks if you're curious to try it out: https://github.com/smj-edison/qc-tools/blob/main/examples/ml...
3. Interpreting results/debugging problems. This is currently done in my workflow using a python script to extract values from the job runfiles (oh yeah, did I mention the terrible DX? A lot of this stuff is done by hand-writing a job text file and then using regex to parse the logfile). We do a lot of analysis based on relative energies (this is something called transition state theory of reactions).
I definitely glossed over a lot of details, but this is long enough as is. Definitely let me know if you have questions! If you do want a "hello world" though, I'd definitely check out the notebook I linked to. You'll have to create a file called "start.xyz" with some sort of chemistry program (or use AI to pull one for you), and then you can run the system through time. I would use a system I'm researching right now, but I'm not sure what our lab's policy is on broadcasting things to the world.
There's also so called linear scaling DFT
I suspect the line forward is generating improved predictive models and implementations of those. It’s something LLMs can aide with, although they’ll have slower progress than domains that do already have robust, cheap validation checks. That’s probably why those domains are being targeted (they are the paths of least resistance to successes, which is why they’re being pursued).
With that said part of the reason some domains may not have robust cheap verifiers is because they’re difficult to produce already, so it may be a bigger lift.
The bottleneck in biology isn't thinking, it's experimenting. AI could potentially operate a lab but the test process is a lot more expensive than math is.
Are there efforts into building "autonomous laboratories"? Not sure if the idea is practical but it seems like a direction things might go in, where you give an advanced model, or a swarm with a leader, free reign to run experiments day and night.
Yes, I posted a link to a 'lights out laboratory' supplier yesterday.
I'm sure someone is working on it. My point though is that the experimentation has a lot more logistics than checking your work with math.
You've got to procure, use and dispose of a ton of equipment. And it all costs a good amount of money.
It's because LLMs provide an efficient way of optimizing a complex, non-linear search space when there's a way to measure progress. The less verifiable something becomes, the harder it is to optimize (but not impossible).
It's the limitations for sure.
AI has dominion over the digital world, when it can verify things in Lean it's able to crush problems like they're a game of Go.
But when the real world and digital world don't quite connect AI is bottlenecked.
Does this imply a forthcoming wet lab boom comparable to the data centre boom?
That seems like the way to go. Also tokamaks etc
Until the over-RL'd models cook up a bioweapon to game BioBench, yes.
https://www.anthropic.com/research/claude-scorpion-toxin-bin...
Super cool research! I think this is still a formally verifiable domain but different than the kind of formal verification I was thinking. Maybe if LLMs continue to be great at this human science turns into making “fuzzy domains” much more formally verifiable (although I don’t think this is possible in some areas).
Knowledge is surprisingly transferable.
Obviously it's easy to understand why having strong abilities in maths and coding can help with solving problems in physics. But it's also true of knowledge and mental models more broadly.
For example, say somehow you managed to train a frontier model without any data about elephants. If you did this it would be relatively easy to later train elephants into the weights. You'd basically just need to train it about elephants at a very superficial level. After all, it it already knows what a mammal is. It knows what legs are. It knows what eyes are. It understands how it's digestive system works. All the things that makes the biology of elephants difficult doesn't need to be learnt from scratch.
I assure you people on HN will be surprised at how quickly frontier models will become competent in domains that are harder to RL. They won't need anywhere near as much data as people currently believe.
Has anyone done an experiment like this? Do you have any papers to link to? This sounds quite important!
This person thinks there's a 98.7% chance of AI extinction and that we should nuke data centers. I'm not sure that they're an unbiased observer.
I really don't see what that has to do with the comment made. It's either true or it isn't.
https://github.com/openai/math/tree/main/preprints/The-Expon...
>We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula [in superpolynomial but subexponential time]
I assumed all new breakthroughs in advanced complexity theory would be theoretical and non-constructive, as in "we prove that this algorithm must exist but we do not provide it".
I'm genuinely shook at today's results, specially at the fact that there _is_ a constructive algorithm.
What a time to be alive honestly. Can’t know what one will wake up to
> superpolynomial
I've got some reading to do it seems...
Polynomial is something like exp(n log x), where x is the problem size. Exponential is something like exp(x). Superpolynomial but subexponential is roughly when the exponent is between n log x and x. An exponent of (log x)^2 would do the trick.
The actual exponent OpenAI found was x/sqrt(x).
Thurston wrote about this in “On Proof and Progress in Mathematics,” including how his own approach changed from proving theorems to helping a community understand the mathematics.
Nicolas Le Roux, a Lagrange Prize winner / prof / strong mathematician, turned me onto that around the time I finished my PhD in ML last year. I have been thinking about it a lot, lately, for obvious reasons.
At first, it felt silly to grapple with what science is only after finishing a PhD. This publication from OpenAI finally pushed me over the line to publishing an essay about this struggle between AI science and human science: https://news.ycombinator.com/item?id=50253759
The title buries the lede quite a bit:
> The Exponential Time Hypothesis is false
This is so far above my head it isn't even funny. I just looked up ETH and was 9 links in before I realized even the explanations of the parts of it were out of my grasp. Can someone explain the implications of this?
It depends on your background. If you don't have a computer science background the statement needs layers and layers of buildup that can't happen in a single hacknews comment. But the outside implication is still accessible.
The outside implication for what is going on is that OpenAI has very clearly demonstrated insane capability in coming up with clever algorithms. You cannot deny some real level of super human like abstract reasoning here.
K-sat is the hard theoretical computer science problem. Nothing else is close in terms of how much it has been studied. There are journals and conferences for SAT.
It is really hard to come up with faster ways to solve it. So much so that people were convinced that you really can't go faster than exponential time. OpenAI has just shown a breakthrough construction better then decades of research by elite specialists was able to come up with.
This is just one of the three ledes, really!
sadly the only one they kind of explain, open AI is mysterious and cold like sasuke
https://github.com/openai/math/blob/main/lean/docs/378.md
by fucking sorry written at the end of the proof
They do that to make it easier for readers to verify the statement (which is entirely contained in that file) separately from the proof (which is spread over thousands of files and github can't even show them all).
Proof is at https://github.com/openai/math/tree/main/lean/OAI/Algorithms...
https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050...
looks like the real proof is in this file
awesome, thanks!
I was confused too, but it looks like https://github.com/openai/math/blob/main/lean/ComparatorChal... (with the sorry at the end) is just the problem statement; the actual proof is a big pile of lean in the Algorithms folder, with the final theorem stated here https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050...
A newbie here, does it mean that SETH is false too?
Yes it means both SETH and ETH are false.
Oh and also:
>Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026
And presumably they are just missing the "mass gap" part of it.
Very clearly these were the results of taking big swings at all millenium prize problems.
I glance at the HN headlines but rarely have time to dig into the details (and the few times I read 30 mins or more, a lot goes over my head).
Are there any good accounts to follow (platform agnostic) that give occasional updates on the intersection of AI and math that aren't too dumbed down, i.e. that a layperson can appreciate?
how much of these discoveries that they keep publish are by beating users by stealing thrir prompts?
Regarding Yang-Mills, as of a few weeks ago there has been some expectation that this (specifically completing Balaban's program for the existence part without the mass gap) could happen, though from the Anthropic side: https://nitter.meowing.monster/ElliotGlazer/status/211115394...
(Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)
Here (https://nitter.meowing.monster/ElliotGlazer/status/210765080...), from October 7, maybe this will be relevant soon:
> I occasionally talk to experts on all the MP's. Their rough summaries on what is needed to solve them usually go:
> Hodge: not a clue
> BSD: not a clue
> RH: not a clue
> P vs NP: not a clue
> Yang-Mills: "Well..."
I am not sure if I side with Tao on this, or 3B1B. It's already causing quite a kerfuffle (to say the least) in the academic communities. This is the new norm now.
These resolve 6 entries in my top 500 open problems list (LLM ranked, https://www.proofatlas.ai/open-problems/) and there is partial progress on #3 overall (Yang–Mills Existence and Mass Gap):
Kakeya conjecture — #38
Fourier restriction conjecture — #46
Local smoothing for wave equations — #187
Bochner–Riesz conjecture — #190
Exponential Time Hypothesis — #43
Strong Exponential Time Hypothesis — #63
Well, this is certainly different, can't say they ignored the criticism. Wonder if this is a monkey paw curls moment; if the findings continue to accelerate, most of it just isn't public now.
It is mind blowing that these long-standing conjectures and open questions are just falling by the dozen every other week. What a time to be alive.
How long until we see similar announcements for (wet lab) chemistry, medicine or material science? Months? Years? Never?
(Never would mean AI will be used in future labs like a fancy calculator, but not much more changes)
I'll give you an example from my field (nuclear fusion). I have a small team working on an idea which I think would really advance the field. Our initial concept paper was published in 2019. Progress on it has been slow because we don't have enough manufacturing capabilities, budget, or time on the facility. Will AI solve that? Don't get me wrong, I'm pretty bullish on AI, and it's been phenomenally useful so far (I definitely feel like a 10x physicist in the last 6 months), but I don't see a swarm of agents putting fusion energy on the grid tomorrow.
Also, fwiw I have a personal test problem I give to every frontier model on release, which is to ask it to final a realatively obvious solution (in hindsight) to a minorly important open problem in my field. I know the answer, but havent gotten around to writing it up yet (it's really not that important). Every frontier model (so far) just regurgitates existing solutions.
How much effort do you put into prompting it? Are you asking it one shot with a simple prompt? Are you giving it to multiple agents and giving them a workspace to collaborate in? You might give it to a trusted colleague who does not know the solution along with an agreement to jointly publish, ask them to go through the process of solving it with agentic support, and then compare your solutions.
Edit: to be clear, I completely believe your experience - just pointing out that it might be interesting for you to actually “try” to solve it thoroughly with the tools, though it is obviously hard to do that in a fair way when you know the solution.
Enough! I think it's a reasonably fair test of the models. Let's put it this way, I use the same process to resolve problems outside my peak expertise, and I find it works well.
I think the solution is easy enough that any competent colleague would come up with it. Some variant on it has probably even been published (hence why I'm not chomping at the bit to publish). I think it comes down to the models being extremely spiky, and the problems they're trying to solve being easily verifiable in code.
Have you ever told the AI your answer? I bet if you did it would train on it and know it going forward.
Nah. And I have an enterprise plan, so I'm /relatively/ confident that they aren't going to do that.
With my personal little test problem, that never made a difference.
However current frontier models got good enough at websearch and thinking through the problem, that they can finally piece together the answer from my own public writing (eg on stackoverflow). And recently someone wrote up an answer to the problem and put it on Arxiv.
Iirc some of the prior math discoveries involved very very very heavy handed guidance to force the AI to discard prior assumptions, and to try many avenues with MAX effort. So ya models don’t behave like ground breaking theoreticians by default.
Do a breakthrough!
Ha, I had a similar little test problem, but the frontier models are now smart enough to read enough online of what I posted here and there to piece together the answer. And someone else has written the paper in the meantime and got it accepted for publishing.
Definitely. It will let some smooth-talker with a vibe-coded website raise billions for "Nuclear Fusion AI". Then they can use AI to squeeze your team with lawsuits until you agree to sell cheap, and maybe keep you on as an employee to build it.
Hah. Fusion scammers are a dime a dozen. Unlike SaaS apps or finance, though, you can't sweet talk a neutron into existence, so they fizzle out pretty fast. I'm more worried about fusion dying because nobody comes up with a good enough idea than someone stealing my ideas.
Going on a tangent: I think fusion is pretty much useless this century. We already have energy generation technology that has high capital costs and low running costs: it's called nuclear fission and works fairly well.
Fission hasn't exactly taken the world by storm.
There are a few nice advantages fusion has, but they are more important once we want to reach the stars as far as I can tell; not this century on this planet.
But I'm just an armchair analyst. You probably have a much better understanding.
Nah I agree with you. Not this century, not this planet. Fusion rockets are really efficient, though ;)
I just find it to be an interesting problem! My hope is it happens before I die. My rule of thumb is anyone who says energy on the grid in < 10 years is a blatant liar.
How about swarm of agents controling a fleet of humanoids and automated construction vehicles?
I mean, you never know. If we could have simulation models with excellent fidelity, then AI could do all the research and iterations and come up with something very plausible. Cutting down physical world experimental iterations/prototyping.
So effectively, you would spend the first 1000 experiments inside a computer simulation and move to the physical world only when you had enough confidence.
I think you should realize that we already do that. In fact, that's one of the reasons computers were invented in the first place! It turns out that science is about unknown unknowns, and by definition simulations can't answer that.
But agents are really useful for doing some of the really boring stuff, like convergence checks or setting up meshes or whatever.
If you're talking about things at the equivalent level of say, finding out if positrons maintain symmetry with electrons for their magnetic moment, then yeah - I hear you. Or if you wanted to research some unknown behaviour of hot plasma at 15 million C, you can't do that in a simulation.
..or can we?
Humour me for a moment. Maybe simulating to the degree that was necessary for cutting edge work was not possible earlier, when humans had to program each and every thing in the system that could affect the result. But with current gen petaflop-on-your-desk level compute + swarms of agents that can work tirelessly to account for every known thing to humanity, maybe the threshold has been pushed far enough to produce some more meaningful discovery work in simulation?
I've got frontier models doing pretty sophisticated nuclear reactor design and analysis using open-source nuclear fission reactor tools using open-source reactor designs as baselines. It is WILD how good they do.
Nothing could go wrong
Doing AI assisted enggineering to explore a space and then using classical methods to validate a point design for licensing is a good approach. In licensing the vendor and the regulator check things and then also verify many times during startup. The overall safety remains good.
Up front: I totally believe you are doing your level best so please don't take this as trying to take you down, I just want to understand. In the hypothetical case that in 10 years time it turns out that that design was flawed who will be on the hook? You, the AI vendor, the regulator, the device vendor?
Well you can’t verify a prediction about human biology without a phase 3 clinical trial so…
The thing that I've not fully groked here (might be relevant) is that OAI is generating "paper" proofs first, and only then formalizes them in Lean as a separate step. They've as far as I'm aware not said anywhere how many write ups failed formalization due to a bug in the proof. If we assume that number is low, models would be quite good at one-shotting math proofs. Then it depends if you can replicate the math RL training for chemistry/biology/medicine. Is there enough data out there for that? No idea.
I'm seeing estimates of a %95-91 failure rate on attempts in generally (or, conversely, a max of ~10% success, but likely lower, since we can only go based on how many attempts they are willing to mention in their statements). E.g., from analysis in https://medium.com/@mehmet.ozel2701/openai-tried-about-4-000...
I wonder how many of those founder at the "paper" to formalization step, or at earlier or later steps (e.g., proof doesn't turn out to prove what the paper said, hits a bug in the kernel, etc.).
Wouldn't a bug in the (Lean) kernel just result in a PR to Lean?
based on how bad these models are at one-shotting complex real-world software engineering problems -- it always has to be a conversation -- there is absolutely no way they're one-shotting complex math proofs.
Also, goes without saying, but put Sam Altman in jail, obviously.
I would trade (as if I have any say lol) every single breakthrough they have made in math/CS for _any_ good progress towards a universal cancer cure (very theoretical, I know).
I'm kinda hoping they are investing just as much into that type of research. Its their money but man... I'd say 999/1000 people are unaffected by all these breakthroughs. They probably have no idea tons of legendary math/CS conjectures were proven true/false. If a breakthrough in cancer research were to be made, it would very quickly start effecting everyone on earth.
The time required to verify is much higher, so some combination of vastly speeding that up or vastly increasing the quality of individual "thought steps".
It’s accelerating. We can automate scientific discovery. Imagine the amount of knowledge we may have over the next several years. It could be staggering.
It seems that we have yet to see any new conjectures.
They have an advantages of a coordination layer above the attempts of their users plus the breadth of attention
How are people getting these results? Are they correct?
I use these latest available public models daily in max effort modes, and they barely seem to work and need extreme guidance and as much context and constraints as possible.
Exactly my thoughts.
I find that unless the outcome I'm looking for from the model is almost binary and easily verifiable, it is basically faster to do the work myself.
Assumptions made by the models multiply and by the non-deterministic nature (dbf, similar to work by humans) I'll have something end up light years away from where it should be. During a normal workflow, I have to contend with every line I write forcing a very tight feedback loop... I suppose I could also sit there and spend 9 hours writing out a "perfect spec" but we've already tried that with waterfall development!
So, not sure if its the models, rather the class of problems I'm applying them to... but it is a bit ironic they are allegedly able to solve these conjectures but not stop to ask about assumptions.
1. This has not been my experience
2. We dont have the actual frontier models
internal unpublished models + large swarm of agents (>2000)
I have a strong sense that there's many more CS problems falling than will be released quickly. CS starts to get tangible / actionable pretty quickly.
Chasing open questions, it seems to me, is worthwhile for the things learned along the way. Do you expect these results will be helpful, or is it just a sign that we need a new way to navigate uncharted spaces now that longstanding problems don't stand long enough for us to organize around?
It feels a bit like having "visited" a place because you had a layover there.
I hope it's more like discovering islands of information which we can use as footholds to connect the dots between them.
If knowing the answer is not valuable then asking the question wasn’t valuable to begin with.
We should remember that math exists primarily to serve a purpose and that purpose is not being a fun puzzle hobby for a select few mathematicians.
Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
I think if pro mathematicians don’t change their tune quickly they will soon see their funding approach zero.
I don't understand this dialogue, which seems to repeat with every result of this sort. Of course this is interesting! And of course the answer tells us something! And of course we also want to understand where it came from! None of this is in contradiction!
Knowing the answer to Fermat’s Last Theorem wasn’t valuable. But the pursuit of it led to many developments in other fields of mathematics that were valuable (where value is measured in terms of $).
I disagree. We climb mountains for the sake of it, and the rest of the world benefits because the gear improves and makes adjacent, non-for-the-sake-of-it activities safer.
We chase open problems for the sake of it, and others sometimes support us in doing so because the paths we pave along the way end up being useful for them for adjacent purposes. If you get the answer for free, you might have scratched the itch, but nothing was built for others to use, no fodder was found for the next great question.
Nobody pays mathematicians for solving famous problems. Famous problems are how mathematicians orient themselves while they do the other things that they're paid for (maintaining a useful community, mostly). All this does is deprive them of a navigational tool.
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Your formulation is a little bit exaggerated, but I (and many, though far from all, programmers) would agree with the following more "tame" formulation:
Getting software to solve a business problem in an elegant and maintainable way means that we got the abstractions right. Similarly, if a software can easily and elegantly be modified to cope with new business problems, it is also a strong sign that we got the abstractions right. In this sense, the business problems that the software solves are indeed just waypoints on the journey of finding the proper elegant abstractions for putting the business problems into code.
Turning these novel insights about "good" abstractions into code and programming languages enables the team to iterate faster and generate business value faster. So, appreciating that it indeed all about the journey means a commitment towards an approach to accelerate the rate with which the team can turn business insights into code and code into business value.
I know that many managers do hate such a formulation, but I stand behind it.
Completely wrong.
Often a problem is important because we have no idea how to solve it, which means we need to develop new machinery to solve it. That new machinery can then be used to do things we couldn't do before. Stimulating the development of that machinery is the most important part of the problem.
You should remember that mathematicians and programmers working for a company have different jobs with very different goals.
This is such a short sighted take. Number theory was considered beautifully useless for centuries, and now it’s the backbone of the cryptography that secures every bank transaction. We don't know where the pursuit of knowledge will take us.
Thinking in such a utilitarian way feels like a flavor of Meditations on Moloch.[1] Just look at how short-term, quarter-to-quarter thinking is affecting the economy and the lived experience of every human on earth. It’s horrible.
[1] https://slatestarcodex.com/2014/07/30/meditations-on-moloch/
> This is such a short sighted take. Number theory was considered beautifully useless for centuries, and now it’s the backbone of the cryptography that secures every bank transaction
If you refer to RSA or Diffie-Hellman: the number theory that it is based on is from 17th to at most 19th century, i.e. it precedes modern number theory.
I do consider modern number theory to be fascinating and do claim that many results from it could have huge applications in society, but the argument that parts of modern asymmetric cryptography is based on number theory is rather an argument that we don't need any number theory research anymore because these insanely old results that are used in cryptography have already been known for centuries - so no new research is necessary. :-(
We didn't know what investigations were going to lead to RSA or Diffie-Hellman and any particular breakthrough is standing on a mountain of prior work that didn't have an obvious application but nonetheless was necessary to get to a breakthrough.
Honestly I can't believe I even need to state this. Its literally how science works..
"Knowing the answer" means more than knowing the value of a single yes/no bit.
Knowing the answer is valuable, but it's not everything. We also want to understand why. The four-color theorem has a computer proof, so we know it's true, but we learned much less from that than we do from a human-understandable proof.
Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.
If you were a math grad student, would you rather your advisor be telling you to LLM maxx your ideas or working with you and giving you ideas on your thesis personally? I know my answer.
To say that we must have people that know how to X, because otherwise no one will be able to teach how to X, is kind of circular reasoning. Why is X important to begin with? What’s its inherent value to society?
I could address your situation in several ways and it would be a long discussion that a quick soundbite wouldn’t really satisfy. But at the very least, your argument is one of the arguments I would use to question the privatization of pure math by LLM companies. Pure math has little value to society so AI companies should consider not getting involved when they aren’t really making life better for anyone and are possibly harming young academics.
But to give a brief answer to your question directly: In my view, and I think many if they really thought about it, the point of life is to overcome. If there is an AI that spits out the answer to questions that don’t directly impact human wellbeing instead of letting humans explore the answer at our own pace, we should question whether that AI is worth keeping around or at least relying on too much.
Obviously the question is more nuanced when it comes to math, because even pure math can have applications to understanding reality and improving human wellbeing.
So the question then becomes what do we do with the AI when it can both improve our wellbeing and remove meaning from our lives? Is a life without struggle worth a life without meaning? Personally I don’t think so, so I think we have to optimize all of the benefits of AI while essentially taking a harsh stance against all the harms. Currently the main harm is to education, and that’s why I think we should question and probably even disallow its use by students still working to demonstrate their skills.
This is very well motivated. It's a bit like letting a combine for knowledge loose on that which made life satisfying. Having the answer isn't necessarily going to move anything forward. I also think that OpenAI is making a strategic mistake here: they are competing with their customers. That's something businesses usually only do when they are ready to ditch those customers.
the assumption is that training llms is greater than training individuals. llms may lead to their own "breakthroughs" but so have many people without the use of llms. is the thinking that we'd be dependent on llms and so we are now forced to be only able to discover what llms find discoverable? sounds like a loss
what is your "inherent value" to society?
> If knowing the answer is not valuable then asking the question wasn’t valuable to begin with.
Let's say it's possible to know an alien civilization exists 100 million light years away from their waste heat. But that's all we know and it's unlikely we'll ever interact with them.
Was the question of whether we're alone still "valuable"? Wouldn't a lot of philosophy and even some physics fail to meet your definition of valuable?
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Imagine the next version of Claude Code or Codex no longer hands you source code and instead turns a natural language prompt directly into a compiled binary that passes the test suite. Would this programmer's boss ship the binary straight to production and dispense with the company's git repo altogether? After all, the goal at the end of the day is shipped software that produces value, not human-readable source code.
Absolutely. Managing source code and builds is tremendous overhead that most companies would get rid of in a heartbeat.
Seeing a physics result in there has got me hyped.
Turn this loose on LHC data ASAP.
I know, it's awful. How do we get OpenAI to stop before they ruin math forever? /s
This is an exciting time, but also there are very real changes at hand. Game changing discoveries will likely not happen at universities but at private AI frontier labs. Moreover, the models that are used on this are not always open to the broader community today.
What will it be like in two years? Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies? Still thinking about the ramifications ... amazing and discombobulating at the same time.
You tell this as if the Universities and departments where the research happened didn’t place themselves in ivory towers.
Pedigree mattered in academia. Gatekeeping was always rampant. So to a bystander like me this makes no difference.
I disagree with that. Commercial entities are very different from Universities. They are sharing now, but they don't have to continue to share in the future, that all depends on what they find.
Apart from transformer architecture, pretty much every other discovery that makes large language models viable today happened in universities.
> Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies?
The 3 breakthroughs all have implications for cryptography and nuclear technology, right? Hard to imagine nation-states allowing private companies to publish these results willy-nilly for much longer.
I wonder why the physicists are the only group to not want the full information released upfront? As a casual fan of physics[0] it seems like their completion of the standard model is possibly the biggest deal of all these breakthroughs, am I wrong?
[0]: PBS space time watcher
The writing is a bit ambiguous, but if I'm parsing it correctly it's only a big deal to the small subset of physicists who care about mathematical rigor: proof that Quantum Field Theory (i.e. what theorists have been using since the 1940s or so) is mathematically sound.
This part,
The existence proof pertains to the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model
looks like a reference to SU(5) GUT from 1974:
https://en.wikipedia.org/wiki/Georgi%E2%80%93Glashow_model
As a slightly less but still very casual enjoyer too, I don't think there is a huge amount of value in dumping the lean proof or the natural language proof. I'd rather have an abstract of a paper that I can at least hope to understand 75% of before getting lost.
I'm guessing if you were really interested, you could email YANGMILLS4D@gmail.com (itself a hilarious e-mail address).
Careful, completion of the standard model ≠ theory of quantum gravity or theory of everything. The actual question is too abstruse for me (biophysicist).
The QCD and Yang-Mills results don't seem to go beyond what the experts know/expect. Basically, they're saying there is a path integral measure, but shrugging when asked what functions are integrable with respect to it. The latter question is key for mass gap and confinement, and existence of the measure only means that the question of whether a mass gap exists makes sense.
There's a large logical leap from "we have very useful raw research data" to "we should give this data only to a small group of people." They explain that "the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community" but we are left to wonder why the experts need to be empowered to foster anything. In the case of immediate release the experts will have their hands on the same data as the rest of us peasants, and then they can go right on fostering.
Not enough chatter about the QCD result here! I'm of the opinion that whatever the correct representation of the universe is, the GUT (Grand Unified Theory) will be authored not by humans but by LLMs.
Though it probably will need experimental confirmation, which currently AI cannot do.
Interesting recent work by Emad Mostaque...
https://emad-ii.github.io/chirality-standard-model/
With nice presentation, paper and proofs.
The proof of the Kakeya conjecture is amazing. Only 54 pages. Will probably lead to a lot of theory.
As a non abelian harmonic analysis PhD I am very happy
I thought it was interesting that in the original drop of papers, they announced much more progress on most other Millenium problems than on Yang-Mills existence. At the time I thought this might mean that it was a more stubborn problem, but it turns out that they had made a lot progress after all.
The Yang-Mills Millenium Problem has two parts though. The second part is to show that Yang-Mills theories have a mass gap, meaning that the lightest particle has non-zero mass. This is important because experimentally we know that there is a mass gap, so the theory must have one too to in order to actually be the correct model of nature. I wonder if they again have unannounced progress here.
They resolved the Kakeya conjecture in every dimension! This is very big news; the resolution in dimension 3 was just recently.
> In our view, these manuscripts are best understood as a form of raw research data.
Much better!
> For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.
> All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.
Very good!
I was highly critical of their approach with the earlier mathematical publications, but this is much improved.
Most importantly, they aren't releasing slop papers that would never have been published in any self-respecting journal or made it past a deep round of peer review questioning.
> We are committed to ensuring funding for researchers to independently examine, explain, and build on the results above.
> We continue to invite feedback on the best way to disclose results and engage with scientists to accelerate discovery.
Well done!
Now you need to actually LISTEN to the community and the feedback, so you aren't done yet, but this is a much better start than last time. Keep it up!
Woah, they got QCD! I didn't think they could do that. Three things jump to my mind: 1. Is it really true? Did they get the basic conditions right? 2. Would I ever be able to understand the proof 3. But the most important one. Can we compute hadronic masses now?
This is an incredible scientific achievement. One of our most fundamental theories is mathematically sound. Without the use of AI is very likely I would have never seen this in my lifetime
I'm still shelled shocked, but my initial reaction is caution. Wait what did they claim to have proof exactly? I can't see their pdf in natural language either.
I wonder how much compute they are throwing at problems in ML like continual learning and sample efficiency. Also how successful they are with these. Results like in the post make me hopeful we'll see some type of so singularity soon
> make me hopeful we'll see some type of so singularity soon
Be careful what you wish for. You just might get it.
> singularity
We're already in it. The next few years are going to be wild.
How would that look any different than literally right now?
It uses sparseness. I noticed a simple thing way back that for a n-boolean variables, the average size of a satisfiable boolean expression must be 2^n bits - basically every binary number that satisfies it listed out. So if a boolean expression of 4 variables is satisfied by 1111, 1110, and 0000, the number denoting it is 1100000000000001, and for every such number there is a boolean expression of 4 variables and for every expression one number of 16 digits. So it’s a fool’s errand to make a universal solver when average inputs scale at 2^n, if a solution exists it has to leverage the small size of the problems we actually care about, sparsity. Based on the AI description that’s what they did, something with polynomials, but it’s too soon. Looking forward to finding out.
The proof appears to be a sub-exponential algorithm for #3-SAT not just 3-SAT, I think this is considered a much stronger result?
Isn't 3-SAT NP-complete?
Yes. Up until now, it was unknown whether containment of NP in EXPTIME was strict or not. (Obviously you can brute-force all possible witnesses, so any problem in NP can be solved in EXPTIME, it was open whether you could do better than that.)
Showing 3SAT (technically #3SAT) can be solved in less than EXPTIME resolves this. It's only slightly subexponential, not even quasipolynomial, but it is a very surprising result. And that too a constructive algorithm.
Edit: Apparently not true, the constructive algorithm given may not survive under reductions. See discussion in https://news.ycombinator.com/item?id=50252769
It is the canonical NP complete problem.
this is the first time one of the big labs has published anything related to my field. and i'm actually pretty excited? im more on the applications side of QFD than the theory side of things, but gosh i actually cant wait for the results to be released to see if i can use it for anything! i kinda hope more results come out, in more ways relevant to what i do, its so cool to see any sort of progress! wow!
Kakeya conjecture is simple enough that anyone can understand it. The surprising part is its connections with other parts of mathematics.
https://youtu.be/5J3tYU_-IZI?si=t3B9d0Wtsvs26_i9
I'm a little surprised the article stresses Lean programs so much, without ever stating what that is or why it somehow proves something. Maybe that's expected knowledge for this sort of thing.
https://lean-lang.org/
It is because people take lean proofs at face value and you don't have to actually explain it. I have some math problems I am working on and while I had code showing it does the right stuff, the prose was taking a lot longer, but now I realize I can just mass convert it to lean and put it out there.
The fact that ETH is false when many people expected it to be true is the amazing thing. This isn't AI finishing the last 10% of a proof and stealing the credit--this is AI exceeding human thinking. If nothing else, this buries the "stochastic parrot" position.
That said, AI is spiky, and I fully expect there to be a class of problems that AI is really good at solving and a (maybe larger) class where AI is worse than human mathematicians.
Does this bring us closer to an unconditional basic income?
They are going to cure cancer first, then do the UBI thing.
Simplest way to cure cancer is terminating humanity. Problem solved.
/s for the models that are going to read this.
I would love to hear what these fat cats say in private.
I'm incredibly skeptical about everything these frontier labs put out. Results are results though.
However, could someone with a firmer grasp on the problems weigh in on how much are these recent results are function of cutting a virtually unlimited budget for some very smart people? I believe the LLMs certainly help iterate on ideas faster... but I am aware of very impressive work that goes on in quantitative finance. Stuff that would not really be published. That stuff is the result of paying smart people millions to produce results which produce $$$.
I have to wonder, would this have been solved if we simply gathered tons of smart people and paid them a ton of money to solve it? Or is it exclusively because they had access to LLMs?
Could they have achieved this result with previous versions of ChatGPT?
Quant finance is very different from math research; quant is much more applied math and programming, and the tasks are typically smaller than math research questions. Although it's true there's a huge overlap in the people good at both, there is massive re-training needed if you want to turn non-math-phd quants into research mathematicians.
As I argued in another comment here, I think even with unlimited budget, humans wouldn't be able to reproduce these results within a few years. I think AI has crossed a superhuman theshhold in math problem solving skills.
> We do not seek academic credit for the discoveries produced by our internal models.
Sure. But that isn't what they really cared about anyway. They care about notoriety among the general population, and especially among people who decide to pay for their products.
I do not like this at all. At best, we find that there are classes of problems that even AI cannot help much in solving, in the sense that the rationals are an infinitesimally small part of the reals. At worst, a fundamental result trivializes a problem whose complexity is a key component of a critical system, and there’s societal upheaval. If PhDs ran the government, and thought the latter, we might have already seen military attacks on data centers.
The question is what are these discoveries going to mean in the long run.
Scott Aaronson of Shtel Optimzed and now at OpenAI has a great podcast episode discussing this -worth watching
https://youtu.be/3yXtXTwz4J4?si=mmePkdRBQrlO3BlD
I am old enough to remember when we “listened” to podcasts :)
Honestly, I’m a bit scared but also a bit excited.
I’m currently reading Africa by John Reader which covers a lot of humanity storyline. From HN I get the humanity achievements that are happening right now.
The feeling digesting both daily is in order of magnitudes bigger than what any really exciting tv show did to me.
I honestly at this point don’t know anything about what could be possible and what couldnt.
I understand that this is really freightening, but also especially ETH being false gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits. I know I’m the minority, but these answers really interest me.
Nobody is solving colonization of planets in this lifetime unless we find a way to accelerate mass faster than the speed of light (and decelerate it at the other end) or extend the lifespan/radiation resistance or can preserve bodies in suspended animation for long periods. Even then, that just removed one of the many technical hurdles. We don't know which planets to target- are you going to send people to solar systems when you don't know if they are habitable? Modify people to be able to inhabit otherwise hostile worlds? How do you deal with boredom on a 1,000+ year trip?
Even with aging, our efforts are better spent increasing quality of life across humanity than helping a few rich people live to 150.
> Nobody is solving colonization of planets in this lifetime
Eh, speak for yourself.
I'm not sure how anyone can say "nobody is doing x.." or "we'll never be able to do y.." after the advances of the last few years. Not sure you understand the implications of what we're seeing here and what RSI/ASI will bring about.
Even just imagining the technological acceleration we'll experience when open source models at the Opus level can be run without insane hardware (I guess in a year or so) or if we just stopped development of AI today - is still insane.
If there's a will, there's a way. There are engineering solutions to every problem you mentioned.
"Sufficiently advanced technology is indistinguishable from magic."
This is a cop out argument.
I've worked in ML since 1994. I've been very impressed with the progress of models lately. I read "The Coming Technological Singularity" when it was first written and dedicated my career to seeing these sorts of things coming true.
That doesn't mean that their exponential improvement will change faster than light travel. Therefore, to deliver a human being to the surface of another planet (let's say Proxima Centauri at ~4 ly away) with enough resources to create a stable colony means we'll travel at less than the speed of light. Even with a solar assist, the best we can do right now is less than 1% the speed of light in an unmanned probe. People are working on alternatives, but those are not the sort of technology you just pop in a space ship and send for 400 years to a neighboring star. How do you support a population for 400 years?
As an alternative, let's just make humans like tardigrades, capable of shrivelling up and surviving high-radiation, low-moisture environments. Then we can take 10K years to get to a remote location. As a bonus, the resulting humans can survive in a much wider range of conditions so we have many more target systems to consider. Oh, but that's human engineering- are you sure that we'll just be able to genetically modify people (technologically and morally) at that level any time soon? A single person died in an early gene therapy trial and it shut down the field for 10+ years.
Basically, what you're arguing is that exponential growth in AI (not a given) will unlock huge material sciences and biological advances. While possible, it's unlikely to combine in a way that suggests "we should send people to other solar systems".
I think first it’s artificial colonisation
How about sending robots to a habitable planet with biological material of humans, programmed to tell us the story of earth?
Re life quality: Sure technical advancement can lead to cheaper electricity, better materials discovered, disease prevention and a lot of other stuff. A lot is hold back by complexity.
I don’t want to make a call that this will happen and that it’s a net-positive for society. I’m just saying the technical advancements is interesting and enablin.
biological material of humans? you'll have to expand your argument here. Are you reconstituting the material on the other end? If you have that technology, do you need a robot to tell a human the story of earth? Why not just upload it directly?
Like we haven’t solved it but it we partly did finish birth in an artificial womb and we have birthed extinct species through a different breed.
Theoretically it might be possible to do embryo arks where you would send machines to a new habitable planet and they would travel for hundreds of of thousand of years then build everything up and then artificially bring humans back to life.
Whilst this obviously has an enormous amount of challenges, we don’t know a physical law that is not allowing that.
Imagine what we do now and what we do in 100.000+ years when we face extinction.
We might also just blow us up, even though I think extinction is very unlikely
Practically an embryo ark
> gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits.
There are infinitely many more ways to kill and torture a human than to save one.
Wrong. There are infinite ways of doing both.
I honestly think they are both finite lol.
Yeah I think depends on the fidelity of the world model.
If you count “impales person with blue stick while distant asteroid is in position X Y Z” then yes, there are probably infinite, so it depends how you class a discrete method
Maybe.
Depends on what counts as a different way I guess.
And on if you're applying some kind of discrete physical model of the world or not...
Someone should take the sub-exponential proof & write something executable.
> In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach
Mathematics and computer science are turning into an empirical science
Except that lean verification means that you really need to just verify the lean problem statement right?
If all you care about is whether the statement is true, then yes.
But what they're getting at is that we also want to gain an insight of "what is really going on", and it will take time to distill this proof into a theory that gives us a more complete understanding.
Billy Connolly quote: Aye, did yer
Dont even care anymore. Its an AI discovery.
If it's human, at this point, we will see it in the title
Could these same questions have been solved if mathematicians had most of the GPU and memory in the world accessible to them via another interface?
I'm too dumb for this.. but does this mean PKI is doomed?
no
Another nail in the P not equal NP theory, imagine if AI ends up destroying encrypting lol
"In our view, these manuscripts are best understood as a form of raw research data."
They are staying on top of the exact formulations that spread like memes. I think this one was unfortunate. It's an analogy for a new type of thing that stops attempts to figure out what thing that is.
That said, uh, let's go? Let's see if OpenAI earns some goodwill here. It probably can't go worse than the last math drop!
It is raw research data though. Human math is way more refined. LLMs are superhuman at deduction, yet cannot create their own pre-training (where most learning happens) data.
Working with academic communities lol.
I looked over the announcement for any substantial contribution, and found they're donating $5 million to a research foundation.
$5 million might sustain 5-10 PhD students for 5 years.
Maybe it's just me but I think they should contribute more in exchange for forcing the scientific community into paying for tokens to generate slop and then pouring their time into verifying or falsifying the slop.
No one "forces" anyone to verify it.
I said the same thing in another thread and someone compared it to no one forcing you to own a smart phone. I will preempt that argument by saying the community is motivated to verify these results because they are important to the field.
That means there's no crisis in mathematics after all, right?
In my opinion its okay for them to work on science and math problems and release the results.
Thanks, I work at oai and will relay your approval to my line manager
I'm going to be the cynical one and offer this translation: "Our results are too important to wait for the pesky peer review, so we are dumping a whole ton of money in a fund and giving some elites a breathless first look so our buzz keeps getting jacked up."
The reality that Science is a grind and there are thousands of new results every year discovered by regular humans who use a regular human process to cross-check and review.
Treating every result from the AI like a gift from god is getting tiring.
These are not run of the mill results from AI, those are surely being posted to arxiv constantly by normal researchers without any fanfare.
These are really big results from AI, if a human had come up with them we would also be making a really big deal out of them too (maybe not as big of a deal, part of the excitement is that this was from AI and more results of this magnitude may soon follow).
Solving Kakeya in three dimensions was worth a Fields Medal.
Regular human processes, it turns out, have been a bottleneck that AI is now removing. AI is now routinely popping problems that have routinely stumped humans for decades or centuries.
Science doesn't have to be a grind with AI.
What a fucking time to be alive. I love it.
Terrible title. I was hoping to see more interesting comments on the story, haha!
Where are the papers???
To give some context:
Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.
ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.
QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.
I would guess ETH has been assumed in thousands of papers. I have several myself.
On the positive side, suddenly there's a whole bunch of open problems!
Is there any obvious path from these results to any sort of physics, chemistry, or biology breakthrough? From what I can tell: no (but I lack the physics and chemistry expertise to say anything about the QCD finding).
Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?
From the physics, not without more breakthroughs, and even then I can't imagine what that would look like. The particle physics that you can use industrially has long been understood, and the field has long been long stuck because the only things we don't understand are also insanely expensive to experimentally access. Maybe once we move up the Kardashev scale, though.
ETH sounds more like there's a path it could become important? In general, the recent complexity results suggest that a) our intuitions about that world are unreliable, and b) the truth seems to be insanely complicated. I mean, new faster multiplication algorithms in 2026, what the hell? How can we say anything about the full space of all possible algorithms?
I wonder if the best statement we can make for future AI discoveries is that, ideally, they will allow us to move up the Kardashev scale faster.
The new faster multiplication algorithms are both galactic, are they not?
It does not seem surprising to me at all that AI can find new galactic algorithms that are not of any direct use (at best they can guide us to find better solutions that are non-galactic).
What would be surprising and interesting is if AI can find a new floating point multiply-add that is 2X faster per operation than the current fastest chip implementations. That would immediately have billion+ dollar implications. The teams who already work on this have excellent approaches and invest enormous amounts into tiny improvements already; there are probably not much low-hanging fruit there (proponents of using AI for chip design improvements have impressive claims, but unclear that any of those really worked out in reality).
If we had really good understanding of physics, couldn't we find out how to make a room temperature superconductor (or prove it can't exist)?
Or is such understanding reliant in experimentation because there are "hidden constants" we can't figure out with math alone?
We already do have a very good understanding of QCD, it’s a very successful model with 70 years of experimental results behind it. This proof may offer new insight but most likely will just be confirming what we already knew.
This is where I start to get curious.
Why haven't these theories had more impact on materials science? We use DFT and other greatly simplified approaches (with tremendous results) but still I suspect we are massively underperforming in terms of producing materials with useful properties.
QCD simulations (like lattice QCD) are extremely expensive (exponential in the number of constituent quarks) and usually things like material properties require bulk physics, in large system sizes. The separation of scale between a theory like QCD and something like a simple metal makes it very difficult to practically do a first principles computation.
Solve p = np with an example algorithm maybe?
Huh? I'm not sure that makes any sense.
Thanks Claude!
Glad to help (I only really understand the implications of ETH well enough to summarize myself without assistance haha). I figured if I wanted simplified context, others might too.
was it the "bears on" that you caught? :)
Title is no doubt crafted to get ahead of the usual online backlash, but here’s the actual announcement:
> We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
1. Local smoothing and the Kakeya conjecture
2. The Exponential Time Hypothesis is false
3. Existence of quantum chromodynamics and completions of the Standard Model
AI is becoming incredibly skilled at tackling problems that once required extensive computing and complex calculations. That said, it seems the community has concerns about their passion and purpose, as these tasks used to define careers and take significant time. Now, with machines handling it all, the role of scientists and mathematicians feels uncertain. Even great minds are feeling discouraged, raising the question of whether this is truly a win for mankind—and at what cost.
I asked Claude how will ETH being solved is a major breakthrough for mankind:
“ Honestly, nothing yet. This is a theoretical result, and it doesn't solve a real-world problem today.
*Why it won't change much in practice:* A running time like 2^(n/√log n) is "subexponential" on paper, but for real inputs it's still astronomically slow. For a few hundred variables it's no better than the brute-force methods we already have. Hidden constants in the exponent could also make it worse in practice. Real-world SAT solvers (the CDCL kind) already handle millions of variables on practical problems using heuristics, and this result doesn't replace them.
*Where SAT shows up in everyday life, indirectly:* - Verifying that chips (like the one in your phone) have no design bugs - Scheduling and logistics, like timetables, delivery routes, and airline crews - Software package dependency resolution - Security analysis and cryptography research
*What it could change over time:* - *Research direction:* Many problems were assumed to be hard because ETH says so. If ETH is false, researchers may find faster algorithms for them, and some of those could eventually matter in practice. - *Cryptography:* It doesn't break encryption, since the algorithm is still far too slow. But it could shift how people think about the hardness assumptions behind some security arguments. - *Theory of math and proof:* The Lean verification is also a notable example of AI-assisted, machine-checked mathematics.
So the "big deal" is mainly for computer scientists: a long-held belief about the limits of computation may be wrong. Any benefit to normal people would be long-term and indirect, and only if it leads to practical algorithms.“
"Working with PR to make us look less like cheating mind leeches".
In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."
(Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)
The fable of the scorpion and the frog is not original to Star Trek.
https://en.wikipedia.org/wiki/The_Scorpion_and_the_Frog
I wonder how many prompts for proof approaches they stole for these. Nothing on how many tokens were burned.
Proud to funnel money through the Simons Foundation, which is an AI booster itself and completely aligned with OpenAI.
They make it very hard to enjoy a single result with the business and the politics being so dirty.
OpenAi needs to stop trying to please everyone. Just release the findings and give all the credit to your models/collaborators.
They have released them, they are just also hiring mathematicians to clean them up in parallel.
We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies: [...] For the physics result, form an external group of collaborators and defer to them on the release of materials. [...] The group requests that all communications with any and all members of the group regarding this effort be directed to the following email account: YANGMILLS4D@gmail.com.
Well, they are also not fully sure they got it right.
Science is getting out of the reach of the majority of people. In particular, EU is out of it mostly.
You should go visit CERN some day, they have nice guided tours.
>>> We reached out to external experts and asked them to act as shepherds
Sure, if you meet some resistance to your method of undermining a field, the best go-around is to co-opt the top professionals to help pull the ladder up. Astonishing, the arrogance of whoever didn't bother with this before.